3

Motion in a
Straight Line
Motion
If an object changes its position with respect to its surroundings with
time, then it is called in motion.

Rest
If an object does not change its position with respect to its
surroundings with time, then it is called at rest.
Rest and motion are relative states. It means an object which is at rest
in one frame of reference can be in motion in another frame of
reference at the same time.
Point Object An object can be considered as a point object, if the distance travelled
by it is very large in comparison to its dimensions.

Types of Motion
1. One Dimensional Motion
If only one out of three coordinates specifying the position of the object
changes with respect to time, then the motion is called one dimensional
motion or rectilinear motion.
For instance, motion of a block in a straight line, motion of a train
along a straight track, a man walking on a level and narrow road and
object falling under gravity etc.
2. Two Dimensional Motion
If only two out of three coordinates specifying the position of the object
change with respect to time, then the motion is called two dimensional
motion.
A circular motion is an instance of two dimensional motion.
 Motion in a Straight Line 19

3. Three Dimensional Motion

If all the three coordinates specifying the position of the object change
with respect to time, then the motion is called three dimensional
motion. A few instances of three dimension at motion are flying bird, a
flying kite, the random motion of gas molecule etc.

Frame of Reference
The most convenient system is a rectangular coordinate system of
three mutually perpendicular axes as X, Y , and Z. The point of
intersection of these three axes is called origin (O ) and considered as
the reference point. The x , y , z-coordinates describe the position of the
object w.r.t the coordinate system. This coordinate system along with a
clock constitutes a frame of reference.

Distance or Path Length Covered

The length of the actual path covered by an object is called the
distance.
It is a scalar quantity and it can never be zero or negative during the
motion of an object. Its SI unit is metre.

Displacement
The shortest distance between the initial and final positions of any
object during motion is called displacement. The displacement of an
object in a given time can be positive, zero or negative.
Displacement, Dx = x2 - x1
where, x1 and x2 are the initial and final positions of object,
respectively.
It is a vector quantity. Its SI unit is metre.

Speed
The time rate of change of position of the object in any direction is
called speed of the object.
Distance travelled ( s)
Speed ( v ) =
Time taken ( t )
Its SI unit is m/s.
It is a scalar quantity.
Its dimensional formula is [M0LT -1 ].
20 Handbook of Physics

Uniform Speed
If an object covers equal distances in equal intervals of time, then its
speed is called uniform speed.

Non-uniform or Variable Speed

If an object covers unequal distances in equal intervals of time and
vice-versa then its speed is called non-uniform or variable speed.

Average Speed
The ratio of the total distance travelled by the object to the total time
taken is called average speed of the object.
Total distance travelled
Average speed =
Total time taken
If a particle travels distances s1 , s2 , s3 , ¼with speeds v1 , v2 , v3 , …, then
s + s2 + s3 + ¼
Average speed = 1
æ s1 s2 s3 ö
çç + + + ¼÷÷
è v1 v2 v3 ø
If particle travels equal distances ( s1 = s2 = s) with velocities v1 and v2,
then
2v1v2
Average speed =
( v1 + v2 )
If a particle travels with speeds v1 , v2 , v3 , ¼ during time intervals
t1 , t2 , t3 , ¼, then
v t + v2t2 + v3 t3 + ¼
Average speed = 1 1
t1 + t2 + t3 + ¼
If particle travels with speeds v1 and v2 for equal time intervals, i.e.
t1 = t2 = t, then
v + v2
Average speed = 1
2
When a body travels equal distance with speeds v1 and v2, the average
speed (v) is the harmonic mean of two speeds, i.e.
2 1 1
= +
v v1 v2

Instantaneous Speed
When an object is travelling with variable speed, then its speed at a
given instant of time is called its instantaneous speed.
Ds ds
Instantaneous speed = lim =
Dt ® 0 D t dt
 Motion in a Straight Line 21

Velocity
The time rate of change of displacement of an object in a particular
direction is called its velocity.
Displacement
Velocity =
Time taken
Its SI unit is m/s.
Its dimensional formula is [M0LT -1 ].
It is a vector quantity, as it has both, the magnitude and direction.
The velocity of an object can be positive, zero or negative.

Uniform Velocity
If an object undergoes equal displacements in equal intervals of time,
then it is said to be moving with a uniform velocity.

Non-uniform or Variable Velocity

If an object undergoes unequal displacements in equal intervals of
time, then it is said to be moving with a non-uniform or variable
velocity.

Average Velocity
The ratio of the total displacement to the total time taken is called
average velocity.
Total displacement
Average velocity =
Total time taken

Instantaneous Velocity
The velocity of a particle at any instant of time is known as
instantaneous velocity.
Dx dx
Instantaneous velocity= lim =
Dt ® 0 D t dt

Relative Velocity
Relative velocity of one object with respect to another object is the time
rate of change of relative position of one object with respect to another
object.
Relative velocity of object A with respect to object B
v AB = v A - v B
If it is in one dimensional motion, we can treat vectors as scalars just
by assigning the positive sign to one direction and negative to others.
22 Handbook of Physics

When two objects are moving in the same direction, then

AB = A - B vA vB
or v AB = v A - vB A B
When two objects are moving in opposite direction, then
AB = A + B vA vB
or v AB = v A + vB A B
When two objects are moving at an angle Q P
q, then
vAB = vA2 + vB2 - 2vA vB cos q vA
vAB b
vB sin q
and tan b = q
v A - vB cos q R vB
– vB O

Acceleration
The rate of change of velocity with time is called acceleration.
Change in velocity ( Dv )
Acceleration (a) =
Time interval ( Dt )
Its SI unit is m/s 2.
Its dimensional formula is [M0 LT -2 ].
It is a vector quantity.
Acceleration can be positive, zero or negative. Positive acceleration
means velocity increasing with time, zero acceleration means
velocity is uniform while negative acceleration (retardation) means
velocity is decreasing with time.

Uniform Acceleration
If an object is moving with uniform acceleration, it means that the
change in velocity is equal for equal interval of time.

Non-uniform Acceleration
If an object is moving with non-uniform acceleration, it means that the
change in velocity is unequal for equal interval of time.

Average Acceleration
If a particle is accelerated for a time t1 with acceleration a1 and for a
time t2 with acceleration a2, then average acceleration
a t + a2t2
aav = 1 1
t1 + t2
 Motion in a Straight Line 23

Instantaneous Acceleration
It is defined as the acceleration of object at any instant of time.
Dv dv
ainst = lim =
Dt ® 0 D t dt

Uniform Motion
If an object is moving along the straight line covers equal distance in
equal interval of time, it is said to be in uniform motion along a
straight line.

Different Graphs of Motion Displacement-Time Graph

S. No. Condition Graph
(a) For a stationary body Displacement

O
Time
(b) Body moving with a constant velocity Displacement

O
Time
(c) Body moving with a constant Displacement
acceleration

O Time

(d) Body moving with a constant Displacement

retardation

O Time
(e) Body moving with infinite velocity. But Displacement
such motion of a body is never B
possible.

O A Time

Note Slope of displacement-time graph gives average velocity.

24 Handbook of Physics

Velocity-Time Graph
S. No. Condition Graph
(a) Moving with a constant Velocity
velocity

O
Time
(b) Moving with a constant Velocity
acceleration having zero initial
velocity

O
Time

(c) Body moving with a constant Velocity

retardation and its initial
velocity is not zero A

B
O
Time

(d) Moving with a constant O

retardation with zero initial
velocity

(e) Moving with increasing Velocity

acceleration

O
Time

(f) Moving with decreasing Velocity

acceleration

O
Time

Note Slope of velocity-time graph gives average acceleration.

 Motion in a Straight Line 25

Acceleration-Time Graph
S. No. Condition Graph
(a) When object is moving with Acceleration
constant acceleration

O Time

(b) When object is moving with Acceleration

constant increasing acceleration

O Time

(c) When object is moving with Acceleration

constant decreasing acceleration

O Time

Equations of Uniformly Accelerated Motion

If a body starts with velocity (u) and after time t its velocity changes to
v, if the uniform acceleration is a and the distance travelled in time t is
s, then the following relations are obtained, which are called equations
of uniformly accelerated motion.
1
(i) v = u + at (ii) s = ut + at 2
2
2 2
(iii) v = u + 2as
(iv) Distance travelled in nth second.
a
sn = u + ( 2n - 1)
2
If a body moves with uniform acceleration and velocity changes from u
to v in a time interval, then the velocity at the mid-point of its path
u 2 + v2
=
2
26 Handbook of Physics

Non-Uniformly Accelerated Motion

When acceleration of particle is not constant then motion is called
non-uniformlly accelerated motion.
For one dimensional motion,
Ds ds
v = lim =
Dt ® 0 D t dt
Dv dv dv d 2s
a = lim = =v× =
Dt ® 0 D t dt ds dt 2
where, Ds is displacement in time Dt, Dv is velocity in time Dt
and a is instantaneous acceleration.
In component form,
$
a = a × $i + a × $j + a × k
x y z

dvx dv y dv
where, ax = , ay = and az = z
dt dt dt

Motion Under Gravity

If an object is falling freely ( u = 0) under gravity, then equations of
motion becomes
1
(i) v = u + gt (ii) h = ut + gt 2 (iii) v 2 = u 2 + 2gh
2
Note If an object is thrown upward then g is replaced by - g in above three equations.
It thus follows that:
u 2h
(i) Time taken to reach maximum height, tA = =
g g
u2
(ii) Maximum height reached by the body, hmax =
2g
(iii) A ball is dropped from a building of height h and it reaches after
t seconds on earth. From the same building if two ball are thrown
(one upwards and other downwards) with the same velocity u and
they reach the earth surface after t1 and t2 seconds respectively,
then
t = t1t2
(iv) When a body is dropped freely from the top of the tower and
another body is projected horizontally from the same point, both
will reach the ground at the same time.