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P
INTRODUCTION
In this chapter we will do motion analysis of moving particles without
taking into account the forces responsible for the motion. We shall study
motion of a particle and get to know terms like position displacement
velocity acceleration and time. These are the parameters by which we
measure motion. We shall study in detail the rectilinear motion and
curvilinear motion of a particle.
Special cases of motion viz. motion under gravity and projectile motion
form an interesting part of this chapter. Study of motion of a particle with
respect to a moving frame of reference would be studied and finally an
alternate solution to rectilinear motion problems using graphical approach
will be dealt with.
RECTILINEAR MOTION
Motion of a particle in a straight line is known as a rectilinear motion. A car
moving on a straight highway lift traveling in a vertical well stone falling
from the top of a building is examples of rectilinear motion.
Position Displacement and Distance
For a moving particle the information of its position at various instants is
a very important data in motion analysis. Position means the location of a
particle with respect to a fixed reference point. Such -fixed point is usually
referred to as the origin ‘o’. The point ‘o’ can be marked anywhere on the
particle’s straight path. In one direction of the origin the position is taken
as positive and therefore the other side of origin implies negative position.
Fig. a Shows a particle occupying position x 5 m while in Fig. b The
particle occupies a position x - 3 m.
Kinematics of Particles

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Displacement is defined as a change in position of the particle. It is a vector
quantity. If a particle occupies position x1 at some time t1 and a new
position x2 at a time t2 then the displacement during the time t2 – t1 is
given by:
AX X1-X2 …………9.1
Displacement is therefore a straight line vector connecting the initial
position to the final position and has no relation with the actual distance
travelled by the particle.
Velocity
‘How fast’ a particle moves is the velocity of the particle in motion. Consider
a particle occupying position x at time t and after a small time interval of At
occupies a new position x + At. ‘How fast’ the particle has moved during the
time interval At is known as the average velocity of the particle.
Displacement
Average velocity
Time
............ 9.2
Av
x x x
t
x
v
t
If the time interval t is made smaller and smaller the average velocity
will become instantaneous velocity i.e. velocity at a particular instant.
Instantaneous velocity is usually referred to as the velocity of the particle
and denoted as v.
0
lim
t
X
V
t
Now by definition
0
lim
t
X
t
is the derivative of x with respect to t.
dx
V
dt
…… 9.2

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S.I Unit of velocity v is metres/sec m/s. The magnitude of velocity is
known as the speed of the particle. A positive value of v indicates that the
particle is moving in the positive direction i.e. the position x increases with
time. A negative value of velocity indicates that the particle is moving in the
negative direction i.e. the position x decreases with time.
For example if vertically upward direction is taken as positive and a stone
is thrown vertically up it will have +ve velocity during its upward motion
till the peak while it will have -ve velocity during its return downwards.
Acceleration
A moving particle has velocity at every instant of its motion. If its velocity
changes the rate of change of velocity with time is the acceleration of the
particle.
If a particle has a velocity ‘v’ at a certain instant and its velocity changes to
vv during a time interval of t the average acceleration of the particle
during this time interval is
v+ Δ v - v
changein velocity
Averageacceleration
time Δt
av
Δv
Δt
a
……9.6
If the time interval t is made smaller and smaller the average
acceleration will become instantaneous acceleration. Instantaneous
acceleration is usually referred to as the acceleration of the particle is and
denoted as a.
0
lim
t
v
a
t
By definition
0
lim
t
v
a
t
is the derivative of v with respect to t
dv
a
dt
…… 9.5
SI unit of acceleration a is m/s
2
. A positive value of acceleration is
indicative of increase in the magnitude of velocity with time i.e. the body is
moving faster in the positive direction. A negative value of acceleration
indicates that the particle is moving more slowly in the +ve direction or
moves more faster in the –ve direction.

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DIFFERENT RECTILINEAR MOTIONS
Different types of rectilinear motions possible are
1 Motion with Uniform Velocity
2 Motion with Uniform Acceleration
3 Motion with Variable Acceleration.
These are further explained in detail.
Uniform Velocity Motion
For a particle whose velocity remains the same throughout the motion is
said to undergo a uniform velocity motion. For example motion of sound a
train traveling during a certain interval at a constant speed packages
moving on a conveyor belt etc. perform uniform velocity motion.
dx
Weknow velocity v
dt
or dxvdt
22
11
xt
xt
2 1 2 1
Integrating dx vdt
x -x v t -t
Here
21
x
xx
distance traveled s since velocity is constant
and
21
t t t
time interval
s vt
s
or v uniform velocityEquation
s
…….9.6
Uniform Acceleration Motion
A particle is said to perform uniform acceleration motion if its velocity
changes at a uniform rate.
If u is the initial velocity of a particle v is the final velocity and t is the time
interval then the acceleration a of the particle is
v-u
a
t
or vu+at uniformAccelaration Equation1 …..9.7 a

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If s is the distance travelled during this time interval then
s Average velocity×t
s
2
2
uv
t
u u at
st
2
1
or uniformAcceleration Equation 2
2
s ut at …..9.7b
From Equation 9.7 a we have
vu
t
a
Substituting in Equation 9.7 b we get
2
22
v-u 1 v-u
s u + a
a 2 a
v -u
s
2a
2
or v u +2 asuniformAcceleration Equation3
Motion Under Gravity M.U.G
Any object projected vertically up in the air or
projected vertically down towards the earth performs
a rectilinear motion with uniform acceleration. The
acceleration is constant and its magnitude is g 9.81
m/s
2
. M. U. G therefore is a special case of uniform
acceleration motion.
Consider a ball thrown vertically up with an initial
velocity v0 from the top of a tower at A and of height
h. The ball would travel vertically up performance
rectilinear motion.
The velocity keeps on reducing at a uniform rate of
9.81 m/s every sec. i.e. 9.81 m/s
2
till it becomes zero
at B. This is the maximum height ymax reached by the
ball. The downward motion of the ball now begins
and the velocity goes on increasing from zero at a
uniform rate of again 9.81 m/s
2

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While passing point A it will have the same speed v0 but pointing
downwards. The ball would finally land on the ground at c with a velocity vc
after time t sec. Knowing v0 and h we can find the maximum height ymax
time of flight t and final velocity of landing vc using equations of uniform
acceleration.
For solving problems on M.U.G follow the guidelines listed below.
1 Take the starting point as the origin and take all directions either +ve
or +ve.
2 By directions we mean the direction of displacement velocity and
acceleration.
3 With proper sign convention use the three equations of uniform
accelerations viz.
2 2 2
1
v u +at s ut + at v u +2as
2
Here
u is the velocity at the start of M.U.G
v is the linear velocity
s is the displacement of the particle
t is the time interval
a is the uniform acceleration due to gravity whose magnitude is
9.81 m/s2

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EXERCISE 1-A
1. The motion of a car during its 90 minutes journey on a straight road was
recorded as below
Travels at 50kmph for the first 20 minutes.
Accelerates uniformly to a maximum speed of 90kmph during the next
10 minutes.
Travels at this maximum speed for the next 4O minutes.
Accelerates uniformly to a stop in the remaining 20 minutes.
1 Acceleration
2 Deceleration
3 total distance traveled
4 average speed during entire motion.
2. Two trains A and B leave the same station on parallel lines. Train A starts
with a uniform acceleration of 0.3 m/s2 and attains a speed of 36kmph
after which it maintains the same velocity. Train B leaves 2 minutes later
with a uniform accelerator of 0.45 m/s2 to attain a maximum speed of
90kmph. When and where will B overtake A.
3. Two elevators in adjoining wells are vertically 60 m apart start from res at
the same instant and approach each other. The up moving elevator travels
up with a uniform acceleration of 0.3 m/s2. The down moving elevator
travels down with an acceleration of 0.15m/s
2
1 When do the elevators cross each other What is the distance traveled by
each of them at this instant.
2 Solve the problem if up moving lift starts 2 sec after start of down moving
elevator.
4. Cars A and B travel on a straight highway. At t 0 car A travelling at
54kmph decelerates at 0.5 mls2 while car B 50 m behind car A travelling
at 36kmph accelerates at 0.8 m/s2. Find the distances and time taken by
the two cars before E overtakes A.
5. A particle starts from rest and accelerates at a constant rate of 0.5 m/s2 for
some time. Thereafter it decelerates at a constant rate of 0.3 m/s2 and
comes to rest. If the particle was in motion for 2 minutes find the
maximum velocity acquired by it.

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6. A block is released from rest from the top of an inclined plane and it takes
5 sec to reach the bottom of the plane. What is the time it takes to travel
one fifth the distance from the top.
7. An athlete running a 100 m run starts from rest and reaches his maximum
speed in a distance of 15 m. He runs the remaining distance with that
velocity and reaches the finish in 96 sec Determine the initial acceleration
and the maximum velocity acquired by the athlete.
8. Track repairs are going on a 4 km length of railway track. The maximum
the train is speed of 108kmph. The speed over the repair track is 3okmph.
If the train approaching the repair track decelerates uniformly from a full
speed of 108kmph to 3Okmph in a distance of 1200 m and after covering
the repair track accelerates uniformly to full speed of 108kmph in a
distance of 1500 m find the time lost due to reduction of the speed in the
repair track.
9. An automobile starts from rest and travels on a straight path al2 mf szfor
some time. After which it decelerates at 1 m/s2 till it comes to a halt. If the
distance covered is 300 m find the maximum velocit5r of the automobile
and the total time of travel.
EXERCISE 1-B
1. A stone is thrown vertically up from the top of the tower 20 m high with a
speed of 15m/s. find
1 Max. height reached by the stone above ground.
2 Velocity with which the stone hits the ground.
3 Total time of flight.
2. A ball is thrown vertically upwards with a velocity 9 m/s from the edge of a
cliff 15 m above the sea level. What is the highest point above sea level
reached How long does it take to hit the water with what velocity does it
hit the water
3. A stone is thrown vertically up from the top of the tower 40 m high with a
velocity of 20 m/s. Three seconds later another stone is thrown vertically
up from the round with a velocity of 30 m/s. Calculate when and where the
two stones will meet from the foot of the tower.

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4. A stone is dropped from the top of a tower. When it has fallen a distance of
10 m another stone is dropped from a point 38 m below the top of the
tower. If both the stones reach the ground at the same time calculate.
1 the height of the tower
2 the velocity of the stones when they reach the ground.
5. Water drips from a tap at a rate of 5 drops per second. The tap is 900 mm
above the basin. When one drop strikes the basin how far is the next drop
above the basin.
6. Water leaks from a ceiling 16 m high at the rate of 5 drops per seconds
between first and second drop when the first drop has just touched the
ground.
7. Drops of water leak from a tap 2 m above the basin at regular intervals. As
the firs: drop strikes the basin the fourth drop starts its fall. What is the
spacing between the drops 1 and 2 drop 2 and 3 drops 3 and 4 at this
instant.
8. A stone released from rest falls freely under gravity. The distance covered
by it in the last second of its motion equals the distance covered in the first
four seconds of its motion. Find the time the stone was in motion.
9. A ball thrown vertically up was found to travel a distance of 5 m during 3d
second of its travel. What is the initial velocity of the ball
10. A halogen gas filled balloon released from the ground travels vertically up
with a constant acceleration of 2 m/s2. Three seconds later a stone is
projected vertically u from the ground to hit the ascending balloon. The
stone just manages to touch th: balloon at the peak of its path. Find the
velocity with which the stone was projected and the height at which the
stone touches the balloon.

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VARIABLE ACCELERATION MOTION
Variable acceleration implies that rate of change of velocity is nc: uniform.
Rectilinear motions are not always uniformly accelerated but more often
undergo variable acceleration. For example suspend a block from a vertical
spring stretch it and then release. The block would oscillate up and down
undergoing variable acceleration motion. The acceleration here is variable
acceleration is proportional to the deformation of the spring.
Variable acceleration motion is usually defined by acceleration written as a
function of time or velocity or position. For the solution of variable
acceleration motion we make use of the basic three differential relations of
velocity and acceleration given below. Two of them have been derived earlier
as equations 9.3 and 9.5.
dx
v
dt
……. 9.4
dv
a
dt
……. 9.5
dv
from 1and 2 a v
dx
.…… 9.6

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EXERCISE 2
1. The rectilinear motion of a particle has its position defined by the relation
x t
3
-7t
2
+20t-10m.Determine
1 Position velocity and acceleration at t 2sec.
2 Minimum velocity and the corresponding time.
2. The motion of a particle moving in straight line is given by a relation.
32
s t -3t +2t+5 wheres is the displacement in metres and t is time in
seconds. Determine
1 Velocity and acceleration after 4 sec.
2 Maximum or minimum velocity and corresponding displacement and
time at which velocity is zero.
3. The position of a particle moving a-long a straight line is defined by the
relation
32
X t -9t +15t+18
where x is expressed in meters and t in seconds.
Determine the time position and acceleration of the particle when its
velocity becomes zero.
4. The car starts from the rest and moves in a straight line such that for a
short time its velocity is defined by
2
v9t +2t
m/s. Where t is in seconds.
Determine its position and acceleration when t 3 sec.
5. Acceleration of a particle is directly proportional to the square of time t
When t 0 particle is at x 24 m. Knowing that at t 6 sec x 96 m and
v 18 mi: find expressions of x and v in terms of t. Also find velocity and
position at t 2 sec.
6. The acceleration of a particle is given by a - kx-3 m/s2. Knowing at x 2
m v O and at x 0.5 m v 3 m/s. Determine a the value of k b the
particles velocity at x1m.
7. The acceleration of an oscillating particle is defined by the relation a-kx
m/s2.Determine
1 The value of k such that v 12m/s at x2m and v0 at x6m
2 Velocity at x4
3 Maximum velocity.

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8. A particle performing rectilinear motion starts from rest from origin and
has acceleration defined by a 25 – v
2
m/s
2
. Determine the time and the
particle’s displacement when v 4 m/s.
9. A particle moving in the + ve x direction has an acceleration a100-4v2
m/s
2
Determine the time interval and displacement of a particle when
speed changes from 1 m/s to 3 m/s.
10. The motion of a particle is defined by the relation a 0.8 t m/s
2
. It is found
that at x 5 m v 12 m/s when t 2 sec. Find position and velocity at t
6 sec.
11. The velocity relation of a rectilinear moving particle is defined as 4 t
2
- 3t-
1 m/s. At t 0 X - 4m. Determine
a The time at which the particle reverses its sense of motion
b At t 3 sec.
1 Acceleration 3 displacement
2 Position 4 Distance traveled.
12. A particle starting from rest moves in a straight line whose acceleration is
given by a 8 - 0.003s
2
where a is in m/s
2
and s is in meters. Determine
1 Velocity of the particle when it has traveled 40m and
2 Distance traveled by the particle when its comes to rest
13. A point moves along a straight line such that its displacement is s 8t2 +
3t where s is in m and t is in seconds. Find the displacement and velocity
at t 4 sec. Also find the distance traveled in the 10th second and the
change in velocity from t 4 sec till t 10 sec.
14. The acceleration of the particle is defined by the relation a 25 – 3x
2
mm/s
2
. The article starts with no initial velocity at the position x O.
a Determine the velocity when x 2mm
b the position when velocity is again zero
c Position where the maximum velocity is maximum and the corresponding
velocity.
15. A particle performing rectilinear motion has its acceleration given by 1 2
- 5 t m/s
2
. At t 4 sec its velocity is 15 m/s and at t 8 sec its position is
60 m. find at t O the particles position velocity and acceleration. Also
find the time when its velocity becomes zero.

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16. A stone is projected from the ground vertically up with an initial velocity of
20 m/s
1 Knowing that the air resistance causes an additional deceleration of
1.01v
2
m/s
2
determine the maximum height reached by stone above the
ground. Hint: a - 9.81 - 0.01v2 m/s
2
2 Solve for maximum height if air resistance is neglected.
17. A particle starting from the origin with an initial velocity u and travelling in
a straight path has an accelerating of 5t + 6 m/s
2
. The position of the
particle at t 2 sec is 1O m. Calculate the particles initial velocity u and its
position at t 4 sec.
18. The acceleration of a vehicle at any instant is given by the expression
2
10
4
a
v
m/s
2
vehicle starts from rest find its velocity w89hen its
displacement is 125 m. Also find its displacement when its velocity is 10
m/s.
19. A particle moves in along a straight line with acceleration
2
-8
a
s
where a is
in m/s
2
and s in meters. When t 1sec s 4m and v 2 m/ s. Determine
acceleration where t 3 sec.

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CURVILINEAR MOTION
A particle which travels on a curved path is said to be performing curvilinear
motion. Figure shows a particle P moving on a curved path. Let us
understand the terms viz. position velocity and acceleration of a particle
performing curvilinear motion.
Position
It is represented by a position vector r which extends
from the origin of the fixed reference axis to the
particle. It describes the location of the particle w.r.t
the origin. as the particle travels along the curved
path the value of F keeps on changing.
Velocity
Let a particle occupying position P move to P as the
particle performs rvilinear motion. Simultaneously
the position vector r also moves to
r .
Let the time interval be t .
The vector joining P and P is the change in position
vector r during the time interval t . The average velocity is
AV
r
V
t
If the time interval is made smaller the point P will come closer to P and
therefore for the limit 0 t the vector r would be
tangent to the path at p. The average velocity now
becomes instantaneous velocity at P
Δ t 0
Δr
lim
Δt
dr
V
dt
……9.9a
Thus as shown in figure b velocity of a particle performing curvilinear
motion is always tangent to the curved path at every moment.

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Acceleration
Since the direction of velocity is continuously changing in curvilinear
motion it results in acceleration being present at every instant.
Let the velocity of the particle at position P be v and at position P be v
then the average acceleration is given by
AV
V
t
a
Making At smaller and smaller results in
instantaneous acceleration.
Δ t 0
ΔV
a lim
Δt
dv
or a
dt
The acceleration vector can be at any angle as shown.
Curvilinear Motion - Rectangular System
Curvilinear motion can be split into motion along x direction y direction
and z direction which can be independently worked as three rectilinear
motions along the x y and z direction. For a curvilinear motion we therefore
write position velocity and acceleration in the vector form as
2 2 2
r Xi +Yj+Zk and magnitude r x +Y +z
…… 9.10 a
2 2 2
x y z x y z
dr
V V i +V j+V k and magnitude V V +V + v
dt
…… 9.10 b
2 2 2
x y z x y z
dv
a a i +a j+a k and magnitude a a +a +a
dt
…… 9.10 c
For a particle moving in the xy plane its rectangular components are
shown.

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Curvilinear Motion - N - T System
In a curvilinear motion the acceleration vector can also be split along the
tangent to the path and normal to the path.
The acceleration vector is therefore expressed as
2
2
nt
or a a +a 9.11
nt
nt
a e + e
aa
An the normal component of acceleration
represents the change in direction c motion and
is always directed towards the centre of
curvature. Its magnitude r. given by
2
n
v
p
a
….. 9.12
Where v is the speed at the instant and p is the radius of curvature
For circular curves p is the radius of the circle.
For curves which are defined as y f x
3
2
2
2
2
dy
1+
dx
Ρ
dy
dx
….. 9.13
at the tangential component of acceleration represents the change in speed
of the particle. The direction of at is positive along velocity vector or
negative opposite to velocity vector depending on whether the speed is
increasing or decreasing.
For Uniform Speed Curvilinear motion at O
For speed changing at uniform rate i.e.
Uniform tangential acceleration curvilinear motion the following relations
hold true
vu +at
…… 9.14 a
2
1
s ut + at
2
…… 9.14 b
22
v u +2as
…… 9.14 c
here ‘u and ‘V are the initial and final speeds during a time interval of ‘t’.
Also ‘s’ is the curved distance traveled by the particle.
For speed changing at varying rate i.e varying tangential acceleration
motion
2
t
ds
dt
a
…… 9.15

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Relation between Rectangular and N - T components of Acceleration
We may sometimes be required to find the N - T components of acceleration
i.e. a and ay knowing the rectangular components of acceleration i.e a and
ar. This can be done using the relation given below which is
3
x y x y
v
Ρ
a v - v a
…… 9.16
From the above relation radius of curvature p can be calculated and hence
a. and hence an and at can be found out.
EXERCISE 3
1. A curvilinear motion is defined by y 8t
3
- 6t m and ax 4t m/s
2
.At t 1
sec
x v
6 m/s. Calculate magnitude of velocity and acceleration at t 3
sec.
2. A particle starts from rest from the position - 5 - 2 4 m. Its acceleration is
defined by
2
a 3t i 12t j+5k k m/s
2
. Find the particles position
displacement velocity and acceleration at t 2 sec.
3. A particle starting from rest at the position 5 6 2 m accelerates
2
a 6t i-24t j+10k m/s
2
. Determine the acceleration velocity position and
displacement of the particle at the end of 2 seconds.
4. A particle performing curvilinear motion has velocity components given as
x v
32 t- 4 m/s and
y v
4 m/s. At t 3 sec it occupied the position 5
12 m. Determine the equation of the path traced by the particle.
5. If x 1 - t and y t2 where x and y are in metres and t is-in seconds
determine x and y components of velocity and acceleration. Also write
equation of the path.
6. A particle moves along a curved path declined by y l/8 x
2
. At any instant
its x coordinate is given by x2t
2
-4t. Determine its velocity and acceleration
at x8m.
7. A particle travels along the path defined by the parabola y 0.5 x
2
. If the x
component of velocity is
x v
5 t m/s determine the distance of particle
from the origin O and the magnitude of acceleration when t 1 sec. At t
0 X 0 and v 0.

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8. A particle P moves in a circular path of 4 m radius. At an instant the speed
is increasing at a rate of 8 m/s
2
and its total acceleration is 10 m/s
2
.
Determine the particles speed at this instant.
9. A racing car traveling on a circular curve ABC of 400 m radius increases its
speed uniformly from 72kmph at A to 108kmph at C over a distance of 300
m along the curve. What was the total acceleration of the car when it was at
B 250 m from A. Also find the total acceleration of the car when it was at A
and at C.
10. A train enters a curve of radius 500 m with a speed of 6Okmph. Determine
magnitude of total deceleration at the instant the brakes are applied so that
the train stops by covering a distance of 4OO m along the curve. Also
determine the time required by the train to come to rest.
11. The motion of a particle is declined by the position vector r 6t i+ 4t2j
where r is in metres and t is in seconds. At the instant when t 3 sec find
i Tangential and normal components of accelerations ii Radius of
curvature.
12. The position of the charged particle moving in a horizontal plane is
measured electronically. This information is fed into a computer which
employs a curve fitting techniques to generate analytical expression for its
position given by
34
r t i+ t j
where i is in metres and t is in seconds. For t l
sec determine i the acceleration of the particle in rectangular components
ii its normal and tangential acceleration and iii the radius of curvature of
the path.
13. A particle travels along a parabolic shaped track y 10 + O.4 x
2
with a
constant speed of 6 m/s. At x 3 m find a components of velocity b
acceleration.
14. A point moves along a path
2
x
y
3
with a constant speed of 8 m/s. What are
the x and y components of its velocity when x 3 m What is the
acceleration of the point at this instant
15. A particle travels along a cubic curve yo.2x
3
.At a position y5.4m its speed
was 5 m/s and decreasing at a rate of 0.8 m/s
2
.Find a At this instant the
particles total acceleration b velocity components along x and y directions.
c time when the particle comes to rest.

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16. A car travels along a depression in a road the equation of depression being
2
x 180y . The speed of the car is constant and equal to 54 km/hr. Find the
acceleration when the car is at the deepest point in the depression. What is
the radius of curvature of the depression
17. An airplane travels along a path such that its acceleration is given by
a 10i +6t j m/s
2
.If the plane starts from rest from the origin determine at
t4sec. a speed of the airplane b radius of curvature of the path c position
of the airplane.
18. A rocket follows a path such that its acceleration is given by a 4i + tj . At
r 0 it starts from rest. At t 10 sec. determine a Speed of the rocket b
Radius of curvature of its path c an and at components of acceleration. d
Position of the rocket.
19. A particle moves in the x-y plane with acceleration component
x
-5
a
m/s
2
and av 2 m/s
2
. If at t O its velocity is 1O m/s directed at 36.87
0
with
the + ve axis find the radius of curvature at t 8 sec and the
corresponding normal and tangential components of acceleration.
20. A particle is moving in x-y plane and its position is defined by
23
32
r t i + t j
23
Find radius or curvature when t 2 sec.
21. a A. particle moves in x-y plane and its position r 6t i + 3t – 4t
2
j m.
Find the radius of curvature of its path and an and at components of
acceleration when it crosses the x axis again. b Solve for the same
conditions given i 3 t i + 4 t- 3 t
2
j m.

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PROJECTILE MOTION
A particle freely projected in the air in any direction other than vertical
follows a curved path and this motion is referred to as a projectile motion.
The path traced by the projectile is known as its trajectory and is parabolic
in nature.
Projectile motion is a curvilinear motion and can be worked using
rectangular system i.e. splitting the motion along horizontal direction and
vertical direction Since gravitational force acts in the vertical direction it is
a uniform accelerated motion in the vertical direction and a uniform
velocity motion in the horizontal direction assuming the air resistance to be
negligible.
Figure shows a projectile fired with an initial velocity vo at an angle 0 with
the horizontal from the top of the tower at A and of height h. The projectile
travels along a parabolic trajectory and reaches the peak at B. Fig. At the
peak the vertical components of velocity vBy 0. The downward motion now
begins and it finally lands with a velocity vc at C on the ground.
Procedure to solve projectile problems Refer Fig.
step 1: Draw a kinematic diagram showing kinematic parameters initial
Velocity angle of projection range time of flight velocity of landing
vertical displacement given or asked in the problem.
step 2: Resolve the initial velocity
o
v into components
ox
v and
oy
v . Resolve
landing velocity
cy
v. into
cx
v and
cy
v.

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step 3: The curvilinear motion is split into horizontal motion HM and
Vertical motion VM. Make a table with two columns as shown. The
left column HM lists the kinematic terms like v s and t for
horizontal motion with uniform velocity. The right column VM lists
the terms u v s a and t for vertical motion with uniform
acceleration of a 9.81 m/s
. Take a sign convention + ve or
+ ve for vertical motion. A sign convention t + ve means all vectors
like velocity u v acceleration a and displacement s acting
upwards are + ve.
step 4: For HM use kinematic relation v s/t since horizontal motion is
with uniform velocity. For horizontal motion
ox Bx cx
V v v
For VM use kinematic relations
2
22
v u +at
1
s ut + at
2
v u + 2as

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EXERCISE 4
1. A gunman fires a bullet with a velocity of 100 m/s 50
0
upwards from the
top of hill 30b m high to hit a bird. The bullet misses its target and finally
lands on the ground. Calculate a the maximum height reached by the
bullet above the ground b total time of flight c horizontal range of the
bu1let d velocity with which the bullet hits the ground.
2. A ball is projected upwards from the top of a
tower 3O m high and strikes the ground after 8
sec at a point 3OO m from the foot of the tower.
Determine the velocity of projection and also the
maximum height attained by the ball above the
ground.
3. A stunt motorcyclist has to clear a ditch. Find the minimum speed it
should have at A to clear the ditch. Also find his speed at B and the angle
of ramp which should be provided.
4. A ball dropped vertically on an inclined plane at A
rebounds with a velocity vo perpendicular to the
plane and lands again on the plane at B. Knowing
vo 15 m/s determine the range R.
5. A ball is projected from the top of a 2O m
high building with a velocity of 10 m/s at
3O
o
upward with the horizontal. The ball
lands on a sloping ground. Find the
range R and the time of flight of the bal1.
6. An object is projected so that it just clears two
obstacles each 7.5 m high which are situated
50 m from each other. If the time of passing
between two obstacles is 2.5 seconds
calculate the complete range of projection and
initial velocity of projection.

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7. A helicopter moving horizontally at an altitude of
100 m drops a packet. The packet landed on the
ground 250 m away from the point on the ground
directly below the point of drop. What is the
velocity of the helicopter
8. An artillery gun can fire a shell with a velocity of 26O m/s. How much
maximum it can penetrate into the enemys territory as it targets the enemy
from the border.
9. A ball thrown with speed of 12 m./s at an angle of 60
o
with a building
strikes the ground 11.3 m horizontally from the foot of the building as
shown. Determine the height of the building.
10. A ball is projected from the top of a tower 110 m height with the velocity of
1OO m/s at an angle of 250 to the horizontal. Neglecting the air resistance
find 1The maximum height the ball will rise from the ground. 2 The
horizontal distance it will travel before it strikes the ground. 3 The velocity
with which it strikes the ground.
11. A ball bearing slides down a 30
0
plane and leaves the plane with a certain
velocity v
o
. Find for what angle of value of vo the ball bearing ran pass
through a 0.8 m wide slot.
12. A gardener holds a water hose at A
causing the water to fall on the plants at
B. find a the discharge velocity vo of water
at A. b velocity of the water as it falls on
the plants at B.

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13. A fighter plane flying horizontally at
324kmph at an altitude of 35OO m
plans to strike an enemy train moving
with a constant velocity of 9O km/hr.
Find the distance the plane should be
behind the train just as it drops the bomb.
14. An aeroplane is flying in horizontal direction with a velocity of 540 km/hr
and at height of 22OOm. When it is vertically
above the point A on the ground an object is
dropped from it. The object strikes the ground
at point B. Calculate the distance AB ignore air
resistance. Also find velocity at B and time
taken to reach B.
15. A missile M is fired from a certain position A with a velocity of 100 m/s at
an angle of 55
0
upwards with the horizontal. After some time an antimissile
N is fired from the same point A with a velocity of 500 m/s at an angle of
40
0
upwards to the horizontal to destroy the missile M. Find a the time of
flight of missile M before it is destroyed. b the horizontal and vertical
distance from A where the destruction takes place.
16. A box dropped from a helicopter moving horizontally with a constant
velocity vH takes 12 sec to reach the I ground and strike the ground at an
angle of 60o. Determine a the altitude h of the helicopter b the horizontal
distance R traveled by the box c the velocity vH of the helicopter.
17. Water being discharged from a horizontal pipe fixed at a height of 3 m from
the ground falls at a horizontal distance of 4.5 m from the point of
discharge. What is the velocity of water discharge from the pipe.

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18. A ball thrown horizontally from the top of a 5Om high building hits the
horizontal ground 20m from the base of the building. What is the initial
velocity of the ball
19. Find the initial velocity and the corresponding angle of projection of a
projectile such that when projected from the ground it just clears a wall 4.5
m high at a horizontal distance of 6 m and finally lands on the ground at a
distance of 35 m beyond the wall.
20. In an Asian games shot put event a ball of mass 6 kg was thrown from an
elevation of 1.85 m at an inclination of 42
0
with the horizontal. The
horizontal distance covered on the ground was measured to be 21.64 m.
Find a the velocity of projection b The time of flight of the ball c The
kinetic energy of the ball at the instant projection and also at the instant it
touches the ground.
21. A projectile is projected from the ground with a velocity of 25m/s. Find a
The maximum horizontal range. b What is the percentage increase in the
maximum horizontal range if it’s initial velocity is increased by 20.
22. A gunman standing on the ground fires his gun to hit a bird flying at an
altitude of 20m from the ground. The angle of projection being 60
0
upwards
with horizontal. If the bullet hits the bird 2.5 sec after firing find aThe
velocity of the bullet as it left the gun. b The bird is killed instantly. Find
the time it takes to reach the ground. Neglect the height of gunman.
23. A ball is projected on an incline from A with velocity vo at an angle as
shown. The ball lands at C 80 m away from A. If the maximum height
covered above A is 22 m find vo and .
24. A ball rebounds at A and strikes the incline
plane at point B at a distance 76 m as
shown in figure. If the ball rises to
maximum height h 19 m above the point
of projection compute the initial velocity
and the angle of projection .

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25. Ball bearings leave the horizontal trough with a horizontal velocity v0 to fall
through a gap of 60 mm. calculate the permissible range of velocity v0 that
will enable the balls to enter the gap.
26. A ball is thrown upward from a high cliff as shown. With a velocity of 6O
m/s at The ball strikes the inclined ground at right angles. If inclination of
ground is 30
0
as shown determine i Time after which the ball strikes the
ground. ii Velocity with which it strikes the ground.iii co-ordinates x y
of a point of strike w.r.t. point of projection.
27. A jet of water discharging from a nozzle hits a vertical screen placed at a
distance of 8m from the nozzle at a height of 3m. When the screen is shifted
by 4m further away from nozzle the jet hits the- screen again at the same
point. Find the "angle of projection and velocity of projection of the jet at
the nozzle.
28. A boy standing at L2 m in front of a wall attempts to strike a ball at the
height h on the wall. Taking vo 15 m/s and knowing h 7.5 m find the
angle 0 for which the ball strikes the wall at B.

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GRAPHICAL ANALYSIS
The motion of a particle along a straight path can be represented by motion
curves. Common motion curves are position-time x - t velocity-time. v - t
acceleration-time a - t and velocity-position v - x curves.
1 position - time x – t curve
This is drawn with position on the ordinate
and time on abscissa. Since
dx
V
dt
at any
instant of time the slope of x - t curve gives the
velocity of the particle at that instant.
V slope x -t curve
…… 9.17
2 velocity - time v – t curve
This is drawn with velocity on the ordinate
and time on abscissa Since
dv
V
dt
At any instant of time the slope of v – t
curve gives the acceleration of the particle
at that instant.
a slope v-t curve
……
9.18
dx
now V
dt
dx vdt
or dx vdt
Let the particles position be xi at time
i
t and its position change to
f
x
at time
f t
If an elemental strip of width dt is taken between
i t
and
f t
then the
area of this strip is vdt.
vdt
represents the entire area under the v - t curve between
i t
and
f t
f
i
x
x
f - i
dx area under v-t curve
x x area under v-t curve
Or
fi
x x + areaunder v-t curve
…… 9.19

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3 acceleration - time a – t curve
This is drawn with acceleration on the
ordinate and time on the abscissa.
dv
We know a
dt
dv adt
or dv adt
Let the particles velocity be
i
v at the
time
i
t and its velocity be
f
v at time
f
t .
If an elemental strip of width dt is taken between
i
t and
f
t then the
area of this strip is adt
f
i
if
v
v
fi
fi
adt rerepresents the entire area under the a - t curve between t and t
dv area under a - t curve
v - v area under a - t curve
or v v + area under a - t curve ....... 9.20
From an a - t curve the Particle position
f
X carl also be known at any
instant
f
t knowing the particles position xi and velocity vi at an prior
instant
i
t using the area moment method formula given below.
f i i G
x x + v t + area undera -t curve t -t
…….9.21here
f i G
t - and
t t t
is the
abscissa of the centroid G of the area under a-t curve.

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4 velocity - Position v - x curve
This is drawn with velocity on the ordinate and position on abscissa.
We know
vdv
a
dx
a v× slope v-x curve ……. 9.22
From v - x curve we can find the particles
velocity v and the corresponding slope at a
given position x and hence using equation
9.22 we can find the particles acceleration.
Let us tabulate the uses of various motion
curves
Standard Motion Curves
No.
Motion
curve
Use Formula
1. v-t Slope of x - t curve gives
velocity
V slope x-t curve
2. a-t a Slope of v t curve gives
acceleration
b Area under v – t change
in position and hence the
new position.
aslope v-t curve
f x
i x
+ area under v - t
curve
3. aarea under a - t curve
gives change in velocity
and hence the new
velocity
b Area under a - t curve
also helps
in finding the Particles
Position
fi
v v + area under a -t curve
f i i
x x + v t +
G
area undera -t curve t -t
4. v-x Slope of v - x curve helps
in finding
the particles acceleration
a slope v-t curve

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We know that rectilinear moving particles either move with uniform velocity
or uniform acceleration or may move with variable acceleration. The general
or standard motion curves viz. a - t v - t X - t for these different
rectilinear motions are given here.
The particles motion may be easily identified if we relate the given motion
curve problem with these standard curves. For example if the particles a - t
curve is parallel to acceleration axis the particle is in uniform acceleration
motion. Similarly if the x - t curve has a cubic equation it indicates variable
acceleration motion.
a Uniform Velocity Motion curves
b uniform Acceleration Motion curves
c variable Acceleration Linear Variation Motion curves

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EXERCISE 5
1. From graph 1 find position and velocity at t 4 1O and 15 sec.
2. Refer graph 2. Find acceleration at t 10 30 45 and 50 sec.
3. From graph 2 find position at t 2030 40 5O and 60 sec knowing xo
0.
4. From graph 3 find position and velocity at t 6 10 and 19 sec knowing
xo 10mv o 5m/s.
5. Figure shows a - t curve for a article
performing rectilinear motion knowing
that at t 0 v 4 m/s and x 20 m
find graphically the velocity and
position of the particle at t 9 sec.
6. The race car starts from rest and travels
along a straight road until it reaches a speed
of 42 m/s in 50seconds s shown by v-t
graph. Determine the distance travelled by
race car in 50 seconds. Draw x-t and a-t
graph.
7. For a particle performing rectilinear
motion the v - t diagram is shown. Draw
a- t and x - t diagrams for the motion if at t
2 sec x 2O m. What is the v - x graph
of a rectilinear moving particle is shown.
Find acceleration of the particle at 2O m 80m and 200 m displacement
during 6 - 10 sec Also find the total distance traveled.

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8. The v - x graph of a rectilinear moving particle is shown. find acceleration
of the particle at 2O m 80m and 200 m.
9. Figure shows an a - t graph for a particle
moving along the x axis. Draw v- t and x
- t graphs and find the speed and
position of the particle at t 9O sec. Also
find the maximum speed attained by the
particle.
10. Figure shows an a - t graph for are rectilinear moving particle. Construct
v- t and x - t graphs for the motion. At t 0 the particle is at x 0 and it
velocity is v3m/s.
11. A car moves along a straight road such that its velocity is described by the
graph shown in figure. For the first 10 seconds the velocity variation is
parabolic and between 1O seconds to 3O seconds the variation is linear.
Construct the s - t and a – t graphs for the time period
0 t 30
Hint: area
under parabolic curve of base a and height
1
h ah
3

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12. The car starts from rest and travels along a straight track such that it
accelerates at a constant rate for 10 sec. and then decelerates at a
constant rate. Draw the v-t and s-t graphs and determine the time t
needed to stop the car. How far has the car travelled
13. Velocity time graph for a particle moving along a straight line is shown.
Draw displacement-time and acceleration-time graphs. Also find the
maximum displacement of the particle.

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RELATIVE MOTION
So far the motion analysis was done from a fixed frame of reference fixed to
the reference However if the motion analysis is undertaken from a moving
frame of reference i.e. observations by a person also in motion then such
analysis comes under relative motion.
Few examples of situations where relative motion analysis is done are
1 A person .in a moving train observes another moving train on a parallel
track.
2 A captain of a moving ship observes the motion of other ships close to it.
3 The pilot of a fighter plane observing a moving target before striking.
Figure shows two particles A and B moving
independent of each other. Let
A
r and
B
r be
their positions defined from a fixed frame of
reference xoy fixed at 0.
If now A observes B then A will find B to be
occupying the position
B
A
r . This position is
measured from a moving reference located at A. From the vector triangle so
formed we may write
B A B
A
r r r
B B A
A
or r r -r Relativeposition relation
….. 9.23
Differentiating the above relation w.r.t time we have
B B A
A
or v v -v Relative velocityrelation
….. 9.24
Further differentiating w.r.t time we get
B B A
A
or a a -a Relativeacceleration relation ….. 9.24
Here
BB
AA
v and a
B
A
r are the relative position velocity and
acceleration of B
w.r.t A whereas
B B B
r v anda are the absolute position velocity and
acceleration of particle B.
In general the absolute motion of any moving particle say B is the sum of
absolute motion of another moving particle say A and the relative motion of
B w.r.t A.

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EXERCISE 6
1. Wedge B travels to the left with acceleration of 2
m/s
2
while block A travels up the slope of the
wedge with an acceleration of 1 m/s
2
relative to
the wedge. Determine the true velocity of the
block at t 5 sec after starting from rest.
2. a A train running at 60 kmph to the right is struck by a stone thrown at
right angles to the train with a speed of 10 m/s. Find the velocity and
direction with which the stone appears to strike the train to a person
sitting at the window in the train. b Solve if velocity of train is 36kmph
and velocity of stone 18kmph.
3. Trains A and B are traveling on parallel tracks in opposite directions.
Velocity of A is thrice the velocity of B. The trains take 20 sec to cross
each other. Determine the velocities of the trains knowing that train A is
260 m long and train B is 300 m long.
4. At t 0 the location of cars A and B are
shown. The cars travel towards the
intersection. Car A has a speed of 10 m/s
and a deceleration of 1.5 m/s2. Car B starts
from the rest and accelerates at 2
m/s2.Determine at t 3 sec the relative
position velocity and acceleration of car B
w.r.t car A.
5. Two ships A and B leave a port at the same time. A travels at 36kmph in
the North- Vest direction while B travels at 50" south of west at 27kmph.
a Determine the relative velocity of B w.r.t A. c After how much time will
the two ships be 120 km apart.
6. a Two ships move from a port at the same time. Ship ‘A has velocity of
3O km/hr and is moving North 3O
o
West while ship ‘B is moving in
South-West direction with a velocity of 40 km/hr. Determine relative
velocity of Aw.r.t. ‘B. Also find the gap between the ships 2 hrs. Later.

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7. Ship A has a velocity of 45kmph due east relative to ship B which in
turn has a velocity of 60kmph at 30
0
South of West relative to ship C.
Find the velocity of A relative to C.
8. A man walking on a straight path at 15kmph finds that the rain is falling
at an angle of 45" to the vertical. As the man increases his speed to
20kmph he finds the rain now falling at 30
o
with the vertical. What is
the true velocity of the rain. Refer Figure.

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DEPENDENT MOTION
Let us first understand independent motion. Vehicles moving on a
highway lifts travelling in parallel vertical shafts ships moving in the sea
etc. are examples of independent motion of particles. Motion of
independent moving particles are related using relative motion equations.
In a dependent motion system the motion of a particle is dependent on
motion of another or several other particles in the system. The particles
forming a dependent motion system are usually connected to each other
by one or more ropes/strings.
In Fig. motion of counter weight W and the lift
C form a dependent motion system. In Fig.
blocks A B and C form a dependent motion
system.
To find a relation between the position
velocities and acceleration of the dependent
particles we have a method known as constant string length methodC S L
M.
C S L M
This method is used to relate the position velocity and acceleration of two
or more particles connected by a common string. This method is based on
the principle "The total length of the connecting string in terms of variable
position: of the various particles connected to it is a constant whatever be
the positions”.
The following steps are used to establish kinematics relations using CSLM
for dependent motion. Let’s take a example of three moving blocks A B
and C whose position velocities and acceleration relations are required to
be found out.
Take a fixed reference axis
perpendicular to the direction of
motion of the moving particles. If the
particles move in the same direction
only one reference axis will do. If
they move in different direction for
every direction a for every direction
a reference axis is required.

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In the given example blocks A and B move in the vertical direction hence
for them we take a horizontal fixed reference 1. Block C moves
horizontally for which-we have taken a vertical fixed reference 2.From the
horizontal reference 1 mark the variable position
A x
and
f i G
t - and
t t t
of
blocks A and B respectively. From the vertical reference 2 mark the
variable position xc of block C.
Measure the length L of the string in terms of variables
A B C
and
x x x
Here
A B C
L + 2 + ±constant
x x x
.........1 Note that some constants which are
added are string portions wrapped over the pulley while some constants
are subtracted like constant a and b which are lengths from the centre of
pulley to the reference axis or moving particles.
correction to equation 1
In this step a negative sign is attached to the variable which decreases
with time during the motion.
In the example taken up say if B was moving down then A would travel
up and C would travel to the left. In the process the variable
B x
increases
with time while variables
A x
and
xC
decrease with time. We therefore
correct equation 1 and get equation 2.
Differentiate the corrected relation 2 w.r.t. time
A B C
0 -V +2V -V
……. 3
The above equation 3 is the relation between the velocities of particles
AB and C.
Differentiate equation 3 again w.r.t time
A B C
0 -a +2a -a
……. 4
The above equation 4 is the relation between the acceleration of particles
A B and C.
Note that the position velocity and acceleration relations developed
through equations 2 3 and 4 are scalar relations relating only the
magnitude since the direction of motion of the particles have already
been accounted by correction to equation 1.

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EXERCISE 7
1. Determine the acceleration of block knowing
A a
2.5 m/s
2
down for the
arrangements shown.
2. Blocks A and B are connected by an inextensible
string as shown. If block B moves to the right
starting from rest with constant acceleration of 3
m/s
2
. a Find the acceleration of block A. b The
velocity of block A at t 8 sec.
3. Knowing velocity of A is 3 m/s to the right and that of B is 4 m/s to the
left determine the velocity of block C.
4. a Knowing aB 3 mf s
2
t determine acceleration
of block A. b Knowing acceleration of block B
w.r.t. A is 9 m/s
2
determine the accelerations
of blocks A and B.
5. Knowing block A has an acceleration of 0.8 m/s
2
determine the acceleration of block B.

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EXERCISE 8
Theory Questions
Q.1. State the different types of Particle Motion.
Q.2. What are the different types of Rectilinear Motions and list the
corresponding equations applicable.
Q.3. What is normal acceleration and tangential acceleration.
Q.4. Derive expression for maximum height and maximum range for a
projectile or horizontal surface.
Q.5. Write a short note on Motion Curves.
Q.6. Draw motion curves for constant velocity case
Q.7. Prove that area under a-t curve gives velocity and area under v-t curve
gives displacement.
UNIVERSITY QUESTIONS
1. During a test the car moves in a straight line such that its velocity is
defined by v - 0-3
2
9 2 tt m/s where T is in seconds. Determine the
position and acceleration when t - 3 sec. Take at t 0 x 0. 5 Marks
2. A man in a balloon is rising with a constant velocity of 5 m/s propels a bail
upwards with a velocity of 2 m/s relative to the balloon. After what time
interval will the ball return to the balloon. 5 Marks
3. A point moves along a curved path y
2
0.4 2 x Atx m its speed is 6 m/s
increasing at
2
3/ ms At this instant find — 5 Marks
i Velocity components along x and y directions ii Its acceleration.

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4. A train leaves station ‘A’ and attains speed at the rate of 4 m/
2
s
for 6 sec
and then 6 m/
2
s till it reaches a velocity of 48 m/s. Further the velocity
remains constant then brakes are applied giving the train a constant
deacceleration stopping it in 6 sec. If the total running time between two
stations is 40 sec. plot a-t v-t and x-t curve. Determine distance between
two stations. 10 Marks
5. A motorist is travelling at 90 kmph when he observes a traffic signal 250 m
ahead of him turns red. The traffic signal is timed to stay red for 12 sec. If
the motorist whishes to pass the signal without stopping just as it turns
green. Determine i The required uniform deceleration of the motor. Ii The
speed of motor as it passes the signal. 4 Marks
6. A particle of mass 1 kg is acted upon by a force F which varies as shown in
fig. If initial velocity of the particle is 10 m/s determine i what is the
maximum velocity attained by the particle. Ii Time when particle will be at
the point of reversal. 6 Marks
7. In Asian games for 100 m event an athlete accelerates uniformly from the
start to his maximum velocity in a distance of 4 m and runs the remaining
distance with that velocity. If the athlete finishes the race in 10.4 sec
determine i his initial acceleration ii his maximum velocity. 6 Marks
8. A ship A travels in the north making an angle of 450 to the West with a
velocity of 18 km/ hr and ship B travels in the East with a velocity of 9
km/hr. Find the relative velocity of B w.r.t. ship A. 4 Marks
9. A curviline motion of a particle is defind by
x
V 25-8tm/s and
2
y48-3t m
At t0 x0 . Find out position velocity and acceleration at t 4 sec.
4 Marks

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42
10. A stone is thrown vertioally upwards and returns to the starting point at
the ground in 6 sec. Find out max. height and initial velocity of stone .
6 Marks
11. For a particle in rectilinear motion
22
α - 0 . 0 5 V m / s at v20m/sx0. Find x
at v 15 m/s and
n
acc atx 50m. 4 Marks
12. Acceleration of a particle moving along a straight line is represented by the
relation
22
a 30-4.5 x m/s . The starts with zero initial velocity at x 0.
Determine a the velocity when x 3 m b the position when the velocity is
again zero c the position when the velocity is maximum. 4 Marks
13. A particle is projected from the top of a tower of height 50m with a velocity
of 20 m/sec at an angle 30 degrees to the horizontal. Determine: 6 Marks
1 Horizontal distance AB it travel from the foot of the tower.
2 The velocity with which it strikes the ground at B
3 Total time taken to reach point B.
14. A sprinter in a 100m race accelerates uniformly for the 35m and then runs
with constant velocity. If the sprinter’s time for the first 35m is 5.4 seconds
determine his time for the race. 4 Marks
15. A gunman fires a bullet with a velocity of 100m/s
0
50 upwards from the top
of a hill 300m high to hit a bird. The bullet misses its target and finally
lands on the ground. Calculate a the maximum height reached by the
bullet above the ground b total time of flight c velocity with which the
bullet hits the ground. 6 Marks
16. A point moves along the path
2
yx /3 with a constant speed of 8m/s. What
are the x and y components of the Velocities when x3. What is the
acceleration of the point when x3 4 Marks

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