CHAPTER - 3

MOTION IN A STRAIGHT LINE

GIST OF LESSON:

➢ Motion is the change in position of an object with time with respect to a reference
point.
➢ If the object size is much smaller than the distance it moves in a reasonable time, then it
is called point object.
➢ Displacement is the measure of change in position of an object with time in a particular
direction.
∆𝑥 = x2 - x1
➢ Velocity is the rate of change in position or displacement of an object with time.
It's Sl unit is m/s.
➢ Speed is the ratio of the path length or the distance covered by an object to the time
taken.
➢ Acceleration is the rate of change of velocity with time.
➢ Kinematics equations for uniformly accelerated motion (Symbols have their usual
meanings)
v=u+at 2. x = x0 + ut + ½ at2 3. v² = u² + 2ax
𝑎
➢ Displacement of a particle in the nth second is given by Snth = u + 2 (2n-1)

➢ Equation of motion under gravity is given by (Symbols have their usual meaning)
1. v = u + (±g)t 2.h = ut + 1/2 (±g) t2 3. v² = u² + 2(±g) h
➢ Stopping distance for a vehicle is given as ds = -u2 / 2a
➢ If a body travels equal distance in equal interval of time along a straight line, then the
body is said to be in uniform motion a straight line.
➢ If a body travels equal distance in unequal intervals of time then it is said to be in non-
uniform motion.
Δ𝑥 Δ𝑣
➢ For a non-uniformly accelerated motion, v = Δ𝑡 and a = Δ𝑡 .
Δ𝑥 𝑥2−𝑥1
➢ Average velocity is given by, Vav = Δ𝑡 = 𝑡2− 𝑡1

➢ Average speed is the total distance travelled divided by the total time taken.
 ➢ Instantaneous speed is the limit of the average speed as the time interval becomes
infinitesimally small or approaches to zero.
∆𝑠 𝑑𝑠
It is given by v = lim =
∆𝑡→0 ∆𝑡 𝑑𝑡

➢ Average acceleration is the ratio of change in velocity of the object to the time interval.
∆𝑣 𝑣2−𝑣1
It is given by. aav = 𝑑𝑡 =
𝑡2−𝑡1

➢ Instantanous acceleration is the acceleration of a body at a certain instant or the

limiting value of average acceleration when time interval tends to zero.
∆𝑣 𝑑𝑣
It is given by a = lim =
∆𝑡→0 ∆𝑡 𝑑𝑡

➢ Relative velocity of object A relative to object B is given by VAB = VA - VB

and velocity of object B relative to object A is given by V BA = VB - VA.

MULTPLE CHOICE QUESTIONS:

1. A person travels along a straight road for half the distance with velocity v1 and the remaining

half distance with velocity v 2 The average velocity is given by

(a) v1v2

v 22
2
(b) v1
v1 + v 2
(c) 2
2v1v 2
(d) v1 + v 2
2 A motor car moving with a uniform speed of 20m / sec comes to stop on the application of brakes

after travelling a distance of 10m Its acceleration is

(a) 20m/ sec (b) −20m/ sec (c) −40 m/ sec (d) +2m/ sec
2 2 2 2

3 Acceleration of a particle changes when

(a) Direction of velocity changes
(b) Magnitude of velocity changes
(c) Both of above
(d) Speed changes
4 Which of the following four statements is false
 (a) A body can have zero velocity and still be accelerated
(b) A body can have a constant velocity and still have a varying speed
(c) A body can have a constant speed and still have a varying velocity
(d) The direction of the velocity of a body can change when its acceleration is constant
5 A particle starts from rest, accelerates at 2 m/s2 for 10s and then goes for constant speed for 30s and
then decelerates at 4 m/s2 till it stops. What is the distance travelled by it
(a) 750 (b) 800 m (c) 700 m (d) 850 m
6 The given graph shows the variation of velocity with displacement. Which one of the graph given
below correctly represents the variation of acceleration with displacement
v
v0

x0 x

a
a
a
a

x
x x
x

(a) (b) (c) (d)

7 A body is thrown vertically up from the ground. It reaches a maximum height of 100m in 5sec. After
what time it will reach the ground from the maximum height position
(a) 1.2 sec (b) 5 sec (c) 10 sec (d) 25 sec
8 For a moving body at any instant of time
(a) If the body is not moving, the acceleration is necessarily zero
(b) If the body is slowing, the retardation is negative
(c) If the body is slowing, the distance is negative
(d) If displacement, velocity and acceleration at that instant are known, we can find the displacement
at any given time in future
9 A ball is dropped on the floor from a height of 10 m. It rebounds to a height of 2.5 m. If the ball is in
contact with the floor for 0.01 sec, the average acceleration during contact is
2 2
(a) 2100 m / sec downwards (b) 2100 m / sec upwards
2 2
(c) 1400 m / sec (d) 700 m / sec
10 An object is projected upwards with a velocity of 100 m/ s . It will strike the ground after
(approximately)
(a) 10 sec (b) 20 sec (c) 15 sec (d) 5 sec
11 The motion of a particle is described by the equation x = a + bt where a = 15 cm and b = 3 cm/s2. Its
2

instantaneous velocity at time 3 sec will be

(a) 36 cm/sec (b) 18 cm/sec (c) 16 cm/sec (d) 32 cm/sec
12 A particle starting from rest travels a distance x in first 2 seconds and a distance y in next two
seconds, then

(a) y = x (b) y = 2x (c) y = 3x (d) y = 4x

13 The acceleration of a moving body can be found from

(a) Area under velocity-time graph

(b) Area under distance-time graph

(c) Slope of the velocity-time graph

(d) Slope of distance-time graph

A particle moves along x-axis as x = 4(t − 2) + a(t − 2) Which of the following is true ?
14 2

(a) The initial velocity of particle is 4

(b) The acceleration of particle is 2a
(c) The particle is at origin at t = 0
(d) None of these
15
Three different objects of masses m1 ,m2 and m3 are allowed to fall from rest and from the same
point 'O' along three different frictionless paths. The speeds of the three objects, on reaching the

ground, will be in the ratio of

1 1 1
: :
(a) m1 : m2 : m3 (b) m1 : 2m2 :3m3 (c) 1 : 1 : 1 (d) m1 m 2 m3

16 An iron ball and a wooden ball of the same radius are released from the same height in vacuum. They
take the same time to reach the ground. The reason for this is
(a) Acceleration due to gravity in vacuum is same irrespective of the size and mass of the body
(b) Acceleration due to gravity in vacuum depends upon the mass of the body
(c) There is no acceleration due to gravity in vacuum
 (d) In vacuum there is a resistance offered to the motion of the body and this resistance depends
upon the mass of the body

17 Figures (i) and (ii) below show the displacement-time graphs of two particles moving along the x-axis.
We can say that

X X

t (i) t (ii)

(a) Both the particles are having a uniformly accelerated motion

(b) Both the particles are having a uniformly retarded motion

(c) Particle (i) is having a uniformly accelerated motion while particle (ii) is having a uniformly

retarded motion

(d) Particle (i) is having a uniformly retarded motion while particle (ii) is having a uniformly

accelerated motion

18 A car starts from rest and moves with uniform acceleration a on a straight road from time t = 0 to t =
T. After that, a constant deceleration brings it to rest. In this process the average speed of the car is
aT 3aT aT
(a) 4 (b) 2 (c) 2 (d) aT
19 Which graph represents the uniform acceleration

s s
s
s

t t t
(a) (b) t
(c) (d)

The displacement of a particle is given by y = a + bt + ct − dt . The initial velocity and acceleration are
20 2 4

respectively

(a) b, − 4d (b) −b,2c (c) b,2c (d) 2c, − 4d

21 From the following displacement-time graph find out the velocity of a moving body
Time (sec)

30o
O
Displacement (meter)
 1 1
(a) 3 m/s (b) 3 m/s (c) 3 m/s (d) 3

22 Acceleration-time graph of a body is shown. The corresponding velocity-time graph of the same body
is
a

v v
v
v

t
t t
(a) (b) t
(c) (d)
23 In the following graph, distance travelled by the body in metres is
Y
15

10
vm/s

0
10 20 30 40 X
Time (s)

(a) 200 (b) 250 (c) 300 (d) 400

24 The displacement of a particle as a function of time is shown in the figure. The figure shows that

20
Displacement

10

0 10 20 30 40
Time in second

(a) The particle starts with certain velocity but the motion is retarded and finally the particle stops

(b) The velocity of the particle is constant throughout

(c) The acceleration of the particle is constant throughout.

(d) The particle starts with constant velocity, then motion is accelerated and finally the particle moves
with another constant velocity
25 The velocity of a body depends on time according to the equation v = 20 + 0.1t . The body is
2

undergoing
(a) Uniform acceleration (b) Uniform retardation
 (c) Non-uniform acceleration (d) Zero acceleration

26 Velocity-time (v-t) graph for a moving object is shown in the figure. Total displacement of the object
during the time interval when there is non-zero acceleration and retardation is

4
 (m/s)
3
2
1
0
10 20 30 40 50 60
t (sec)

(a) 60 m (b) 50 m (c) 30 m (d) 40 m

27 The velocity-time graph of a body moving in a straight line is shown in the figure. The displacement and
distance travelled by the body in 6 sec are respectively
5
4
3
2
V(m/s)

1
0
1 1 2 3 4 5 6
2
t(sec)
3

(a) 8 m, 16 m (b) 16 m, 8 m (c) 16 m, 16 m (d) 8 m, 8 m

28 With what velocity a ball be projected vertically so that the distance covered by it in 5th second is

twice the distance it covers in its 6th second (g = 10m/ s )

2

(a) 58.8 m/s (b) 49 m/s (c) 65 m/s d) 19.6 m/s

29 The ratio of the numerical values of the average velocity and average speed of a body is always

(a) Unity (b) Unity or less (c) Unity or more (d) Less than unity

30 Which of the following is a one dimensional motion

(a) Landing of an aircraft
(b) Earth revolving a round the sun
(c) Motion of wheels of a moving trains
(d) Train running on a straight track
31 A body is released from the top of a tower of height h . It takes tsec to reach the ground.
Where will be the ball after time t / 2 sec
(a) At h / 2 from the ground
(b) At h / 4 from the ground
 (c) Depends upon mass and volume of the body
(d) At 3h / 4 from the ground
32 A body A starts from rest with an acceleration a1 . After 2 seconds, another body B starts from rest

with an acceleration a 2 . If they travel equal distances in the 5th second, after the start of A, then the

ratio a1 : a 2 is equal to
(a) 5:9 (b) 5:7 (c) 9:5 (d) 9:7
33 A balloon starts rising from the ground with an acceleration of 1.25 m/s2. After 8s, a stone is released

from the balloon. The stone will ( g = 10 m/s2)

(a) Reach the ground in 4 second

(b) Begin to move down after being released

(c) Have a displacement of 50 m

(d) Cover a distance of 40 m in reaching the ground

34 A rocket is fired upward from the earth's surface such that it creates an acceleration of 19.6 m/sec2. If
after 5 sec its engine is switched off, the maximum height of the rocket from earth's surface would be

(a) 245 m (b) 490 m (c) 980 m (d) 735 m

35 A very large number of balls are thrown vertically upwards in quick succession in such a way that the
next ball is thrown when the previous one is at the maximum height. If the maximum height is 5m,
−2
the number of ball thrown per minute is (take g = 10ms )
(a) 120 (b) 80 (c) 60 (d) 40

ANSWERS (MULTIPLE CHOICE QUESTIONS):

Q. ANS Q. ANS Q. ANS Q. ANS Q. ANS
1. D 8. D 15. C 22. C 29. B
2. B 9. B 16. A 23. A 30. D
3. C 10. B 17. C 24. A 31. D
4. B 11. B 18. C 25. C 32. A
5. A 12. C 19. A 26. B 33. A
6. A 13. C 20. C 27. A 34. D
7. B 14. B 21. C 28. C 35. C
Explanation:
Q.5. Velocity acquired by body in 10sec
v = 0 + 2 10 = 20m/ s

and distance travelled by it in 10 sec

1
S1 =  2  (10) 2 = 100 m
2

then it moves with constant velocity (20 m/s) for 30 sec

S2 = 20  30 = 600 m
2
After that due to retardation (4m / s ) it stops
v 2 (20) 2
S3 = = = 50m
2a 2  4

Total distance travelled S1 + S2 + S3 = 750m

Q.9. Velocity at the time of striking the floor,
u = 2gh1 = 2  9.8  10 = 14m / s

Velocity with which it rebounds.

v = 2gh 2 = 2  9.8  2.5 = 7 m / s

 Change in velocity v = 7 − (−14) = 21m/ s

v 21
= = = 2100 m / s 2
 Acceleration t 0.01 (upwards)
Q.18. For First part,
u = 0, t = T and acceleration = a
1 1
S1 = 0 + aT 2 = aT 2
 v = 0 + aT = aT and 2 2

For Second part,

u = aT, retardation=a1, v = 0 and time taken = T1 (let)

 0 = u − a1T1  aT = a1T1
u 2 1 a 2T 2
 S2 = =
and from v = u − 2aS2
2 2
2a1 2 a1

 aT 
1
S2 = aT  T1  As a 1 = 
2  T1 

1 2 1
aT + aT  T1
S1 + S2 2 2
vav = =
 T + T1 T + T1
 1
aT (T + T1 )
=2 1
= aT
T + T1 2

Q.31. Let the body after time t / 2 be at x from the top, then
1 t 2 gt 2
x= g =
2 4 8 …(i)
1
h = gt 2
2 …(ii)
h
x=
Eliminate t from (i) and (ii), we get 4

h 3h
=h− =
 Height of the body from the ground 4 4

1 1
h = at 2 =  1.25  (8) 2 = 40 m
Q.33. When the stone is released from the balloon. Its height 2 2 and
velocity
v = at = 1.25  8 = 10 m/ s

Time taken by the stone to reach the ground

v 2gh  10  2  10  40 
t= 1 + 1 + 2  = 1 + 1 + 
g v  10  (10)  
=4 sec

Q.34. Given a = 19.6 m/ s = 2g

2

Resultant velocity of the rocket after 5 sec

v = 2g  5 = 10g m/ s

1
h1 =  2g  25 = 245m
Height achieved after 5 sec, 2

On switching off the engine it goes up to height h 2 where its velocity becomes zero.
0 = (10g)2 − 2gh 2  h 2 = 490m

Total height of rocket = 245 + 490 = 735 m

Q.35. Maximum height of ball = 5 m

u = 2gh = 10 m / s
So velocity of projection 
Time interval between two balls (time of ascent)
u 1
= = 1sec = min
g 60 .
So number of ball thrown per min. = 60
ASSERTION & REASON QUESTIONS:
Directions: Each of these questions contain two statements, Assertion and Reason. Each of these
questions also has four alternative choices, only one of which is the correct answer. You have to
select one of the codes (a), (b), (c) and (d) given below.
(a) Assertion is correct; reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct; reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct; reason is incorrect
(d) Assertion is incorrect, reason is correct
1. Assertion: A body may be accelerated even when it is moving uniformly.
Reason: When direction of motion of the body is changing, the body must have
acceleration.
2. Assertion: Displacement of a body may be zero when distance travelled by it is not zero.
Reason: The displacement is the longest distance between initial and final position.
3. Assertion: For one dimensional motion the angle between acceleration and velocity
must be zero.
Reason: One dimensional motion is always on a straight line
4. Assertion: The position-time graph of a uniform motion, in one dimension of a body
cannot have negative slope.
Reason: In one – dimensional motion the position does not reverse, so it cannot have a
negative slope.
5. Assertion: Velocity-time graph for an object in uniform motion along a straight path is a
straight line parallel to the time axis.
Reason: In uniform motion of an object velocity increases as the square of time elapsed
6. Assertion: The average velocity of the object over an interval of time is either smaller
than or equal to the average speed of the object over the same interval.
Reason: Velocity is a vector quantity and speed is a scalar quantity
7. Assertion: The speedometer of an automobile measures the average speed of the
automobile.
Reason: Average velocity is equal to total displacement per total time taken.
 8. Assertion: A particle starting from rest and moving with uniform acceleration travels a
length of x and 3x in first two and next two-seconds.
Reason: Displacement is directly proportional to velocity.
9. Assertion: A body is momentarily at rest when it reverses the direction.
Reason: A body cannot have acceleration if its velocity is zero at a given instant of time
10. Assertion: The velocity-time graph of a uniformly accelerated motion in one dimension
of a body can have negative slope.
Reason: When the speed of body decreases with time, the position-time graph of the
moving body has negative slope.

ANSWERS (ASSERTION & REASON QUESTIONS)

Question 1 2 3 4 5 6 7 8 9 10

Answer a c d c c a d c c c

CASE STUDY BASED QUESTIONS:

CASE 1 DISTANCE AND DISPLACEMENT
Distance is a scalar quantity that refers to "how much ground an object has covered" during
its motion. Displacement is a vector quantity that refers to "how far out of place an object
is"; it is the object's overall change in position.

Distance is the total path covered by the body and displacement is the shortest path
covered by the body.
(i) What can be said about the displacement of the body if it the particle comes back to
its initial position-
(a)It is zero (b) It cannot be zero
(c)It may or may not be zero (d)It is negative
(ii) An athlete finishes a round of circular track of radius R in 40 sec. What is his
displacement at the end of 2 min 20 sec?
 (a)2R (b)2πR
(c)7πR (d)Zero
(iii) If the displacement of an object is zero, then what can we say about its distance
covered?
(a) It is negative (b) It is must be zero
(c) It cannot be zero (d) It may or may not be zero
(iv) If s represents distance and S represents displacement, then |S|/s is.
(a) > 1 (b) < 1
(c) = 1 (d) ≤ 1
(v) The location of a particle has changed. What can you say about distance and
displacement covered by the particle?
(a)Neither can be zero (b)both may be zero
(c)only one may be zero (d) Neither can be negative
CASE 2 SPEED AND VELOCITY:
Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought
of as the rate at which an object covers distance. A fast-moving object has a high speed and
covers a relatively large distance in a short amount of time. Contrast to this to a slow-moving
object that has a low speed; it covers a relatively small amount of distance in the same
amount of time.
Velocity is a vector quantity that refers to "the rate at which an object changes its position."
Imagine a person moving rapidly - one step forward and one step back - always returning to
the original starting position. While this might result in a frenzy of activity, it would result in
a zero velocity. Because the person always returns to the original position, the motion would
never result in a change in position.

(i) A boy starts from a point A, travels to a point B at a distance of 3 km from A and
returns to A. If he takes two hours to do so, his average speed is-
(a) 3 km/h (b) zero
(c) 2 km/h (d) 1.5 km/h
(ii) A man leaves home for a cycle ride and comes back home after half-an-hour ride
covering a distance of one km. What is the average velocity of the ride?
(a) 10 kms-1 (b) ½ kmh-1
(c) 2 kmh-1 (d) Zero
 (iii) The ratio of the numerical values of the average velocity and average speed of a
body is
(a) unity (b) unity or less than 1
(c) unity or more (d) less than unity
(iv) If Position of a particle is given by x = (4t2 – 8t), then which of the following is true?
(a) Velocity is zero at t = 0
(b) Velocity is zero at t = 1s
(c) Velocity is zero at t = 2s
(d) Velocity is zero at t = 4s
(v) The position of a moving object is given by x= 3t2-2t+1, where x is in meter and t is
in sec. it’s average velocity between t=1s and t=3s will be-
(a)1m/s (b)3m/s
(c)5m/s (d)10m/s

CASE 3 ACCELERATED MOTION:

In mechanics, acceleration is the rate of change of the velocity of an object with respect to
time. Acceleration is a vector quantity.
Uniform acceleration refers to the constant acceleration of a body irrespective to the
function of time. If an object is under constant acceleration and moves on the x-axis plane,
it is known as a uniform accelerated motion on the horizontal plane or dimension.
An object's average acceleration over a period of time is its change in velocity divided by the

duration of the period. Mathematically,

∆𝑣
ā = ∆𝑡

Instantaneous acceleration, meanwhile, is the limit of the average acceleration over

an infinitesimal interval of time. In the terms of calculus, instantaneous acceleration is
the derivative of the velocity vector with respect to time:

When a body is moving with constant acceleration then some relations can be established
between velocity,acceleration,displacement and time.Those relations are called equations
of motion.
 (i) A body starts from rest and travels with uniform acceleration on a straight line. If
its velocity after undergoing displacement of 32 m is 8 m/s, its acceleration is-
(a) 1 m/s² (b) 2 m/s²
(c) 3 m/s² (d) 4 m/s²
(ii) What is time taken by a body which starts from rest and undergoes displacement
16 m with uniform acceleration 2 m/s².
(a) 4 s (b) 3 s
(c) 6 s (d) 8 s
(iii) The position of a moving object along x-axis is given by x=2t3-4t+3, where x is in meter
and t is in sec. What would be it’s acceleration at t=1 sec?
(a)6m/s2 (b)3m/s2
(c)12m/s2 (d)1m/s2
(iv) A body starts from rest and travels with uniform acceleration of 2 m/s². If its
velocity is v after undergoing displacement of 9 m, then v is
(a) 8 m/s (b) 6 m/s
(c) 10 m/s (d) 4 m/s
(v) A body starts from rest and travels with an acceleration of 2 m/s². After t seconds
its velocity is 10 m/s. Then t is-
(a) 10 s (b) 5 s
(c) 20 s (d) 6 s

CASE 4 MOTION UNDER GRAVITY:

An object released near the surface of the Earth is accelerated downward under
the influence of the force of gravity. The magnitude of acceleration due to gravity
is represented by g. If air resistance is neglected, the object is said to be in free fall.
If the height through which the object falls is small compared to the earth’s radius,
g can be taken to be constant, equal to 9.8 m s–2. Free fall is thus a case of motion
with uniform acceleration. We assume that the motion is in y-direction, more
correctly in –y-direction because we choose upward direction as positive. Since the
acceleration due to gravity is always downward, it is in the negative direction and
we have = – g = – 9.8 m/sec2
(i) A stone of mass 0.05kg is thrown vertically upwards. What is the direction and
magnitude of net force on the stone during its upward motion?
 (a) 0.49 vertically downward
(b) 9.8 vertically downwards
(c) 0.49 N vertically upwards
(d) 0.98 N vertically downwards
(ii) Free fall of an object (in vacuum) is a case of motion with –
(a) Uniform Velocity
(b) Uniform acceleration
(c) Variable acceleration
(d) constant momentum
(iii) Three different objects of masses m1, m2, m3 are allowed to fall from rest and from
the same point ‘O’ along three different frictionless paths. The speed of the three
objects, on reaching the ground, will be in the ratio of
(a) m1: m2 : m3 (b) m1 : 2m2 : 3m3
(c) 1:1: 1 (d) 1/m 1: 1/m2: 1/m3
(iv) A cricket ball is thrown up with a speed of 19.6 m/sec. The Maximum height
it can reach is -
(a) 9.8 m (b) 19.6 m
(c) 29.4 m (d) 39.2 m
(v) A ball thrown up under gravity (g=10 m/sec2). Find its velocity after 1 sec at a
height of 10 m –
(a) 5 m/sec2 (b) 5 m/sec
(c) 10 m/sec (d) 15 m/sec

CASE 5 GRAPHS IN STRAIGHT LINE MOTION:

Graphs are often the best way to convey descriptions of real world events in a compact
form. Graphs of motion come in several types depending on which of the kinematic
quantities (time, position, velocity, acceleration) are assigned to which axis.
There are some standard graphs like straight line, parabola, hyperbola, ellipse, circle,
periodic, exponential etc. which are formed on the basis of relation between the
variables.
(i) The displacement -time graph of a moving particle is shown below. The
instantaneous velocity of the particle is negative at the point-
 (a) C (b) D
(c) E (d) F
(ii) The velocity -time graph of a moving particle is shown below. Total displacement
of the particle during the time interval when there is non- zero acceleration and
retardation is-

(a) 50 m (b) 40 m
(c) 60 m (d) 30 m
(iii) Which of the following is not possible for a body in uniform motion?

(a) (b)

(c) Both (a) & (b) (d) None of the above

(iv) Which of the following graphs gives the equation x=v0t+1/2at2 -

(a) (b)
 (c) (d) None of the above

(v) The relation between time(t) and position(x) for a moving object is given by t=2αx 2.
Where α is constant. The shape of graph between t and x will be-
(a)parabola (b)hyperbola
(c)straight line (d)ellipse

ANSWERS (CASE STUDY BASED QUESTIONS):

ANS (i) ANS (ii) ANS (iii) ANS (iv) ANS (v)

CASE 1 a a c d a

CASE 2 a d b b d

CASE 3 a a c b b

CASE 4 a b c b b

CASE 5 c a a b a