BS Physics (Foundation) Series

1. 1
Module # 06
Equations of Motion
First Equation of Motion
Vf = Vi + at
It is the first basic equation of motion. This equation is the
relationship among the time, acceleration, initial velocity and final
velocity. If any three of these are known, then we can calculate
the fourth one.
If an object is moving with uniform acceleration a, its velocity
changes from initial velocity Vi to final velocity Vf in time interval ‘t’
then change in velocity is given by:
ΔV = Vf - Vi
While
Change in Velocity
Average Acceleration = -------------------------------
Time
Since acceleration is uniform, therefore, we write the above
equation as

2. 2
ΔV Vf - Vi
a = --------- = -----------------------
t t
OR
at = Vf - Vi
OR
Vf = Vi + at
Note: In this equation 'a' is substituted by 'g' for the body falling
from a certain height.
Second Equation of Motion
S = Vi t + ½ at2
This is the second equation of motion. It establishes the
relationship between distance, initial velocity, acceleration and
time.
If a body starts with an initial velocity Vi and moves for ‘t’ seconds
with an acceleration 'a' so that its final velocity becomes Vf, then
the average velocity is given by the relation:

3. 3
Vi + Vf
Vav = ----------------
2
Also the total distance covered by the body is
S = Vav x t
Substituting the value of Vav we get,
Vi + Vf
S = (--------------) X t
2
But according to the first equation of motion, we have
Vf = Vi + at
Therefore, the distance covered becomes
Vi + Vi + at 2Vi + at 2Vit + at2
S = (----------------------) X t = (-------------) X t = (-------------)
2 2 2
OR
S = Vit + ½ at2

4. 4
Note: In this equation 'a' is substituted by 'g' for the body falling
from a certain height.
Third Equation of Motion
2aS = Vf
2 – Vi
2
This equation gives the relationship between the initial and final
velocities, the acceleration and the distance covered.
As the total distance covered by the body is
S = Vav x t
But
Vi + Vf
Vav = ----------- So,
2
Vi + Vf
S = (--------------) X t ________ [1]
2
By first equation of motion
Vf = Vi + at OR at = Vf - Vi

5. 5
OR
Vf - Vi
t = ----------
a
Putting the value of t in equation [1] above, we get
Vi + Vf Vf - Vi Vf
2
– Vi
2
S = (--------------------------) X ------------- = --------------------
2 a 2a
2aS = Vf
2
– Vi
2
Note: In this equation 'a' is substituted by 'g' for the body falling
from a certain height.
Equations of Angular Motion
If the rotation of a body about a fixed axis is considered, then, the
angular displacement, angular velocity and angular acceleration
can be dealt with as scalars. The equations of motion of such
rotating bodies with uniform angular acceleration can be obtained
simply by replacing the corresponding quantities of linear motion
of uniformly accelerated bodies. The following table illustrates the
analogous equations in linear and angular motion.

6. 6
Table: Equation of Motion with Constant Acceleration
Linear Motion Rotational Motion
(1) S = vt  = t
(2) vf = vi + at f = i+ t
vf + vi
(3)  v  = --------------
2
f + i
   = --------------
2
(4) vf
2
– vi
2
= 2aS f
2
- i
2
= 2
(5) S = vit + ½ at2
 = it + ½ t2
(6) F = ma  = I
Where,  represents torque and I moment of inertia.