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Equations of motion


EQUATIONS OF MOTION


  • The variable quantities in a uniformly accelerated rectilinear motion are time, speed, distance covered and acceleration. Simple relations exist between these quantities. These relations are expressed in terms of equations called equations of motion.

 

Equation of uniformly Accelerated motion:

  • There are three equations for the motion of bodies that travel with a uniform acceleration.
  1. First equation of motion: The first equation of motion is v = u + at. It gives the velocity acquired by a body in time t.

Let the initial velocity of a body be ‘u’.

It is subjected to a uniform acceleration ‘a’ after time ‘t’. let the final velocity ‘v’.

Now acceleration = .

Or Acceleration =

So,

Or at = v – u

Or  v = u + at

This is the first equation of motion.

It is used to find out the velocity ‘v’ acquired by a body in time ‘t’.     

 

  1. Second Equation of Motion: The second equation of motion is

.

It gives the distance travelled by a body in time  ‘t’.

Let a body has an initial velocity ‘u’

and a uniform acceleration ‘a’ for time ‘t’

So that it’s final velocity is ‘v’.

Average velocity = .

So, Average velocity =

Also, Total distance travelled = Average velocity × Time

[∵ Average velocity = ]

So,

  (by  putting the value of ‘v’ from first equation).

Or

Where s = distance travelled by the body in time  ‘t’

u = Initial velocity of the body

and   a = Acceleration

This is the second equation of motion.

 

  1. Third Equation of Motion:

The third equation of motion is v2 = u2 + 2as

  • It gives the velocity acquired by a body in travelling a distance ‘s’.
  • The third equation of motion can be derived from the first two equation of motion.
  • From the second equation of motion, we have

s = ut  + at2  …. (1)

From the first equation of motion, we have

v = u +at     …. (2)

From (1) and (2)

          at = v – u

Or   …. (3)

Substituting the value of ‘t’ in equation (1),

This is the third equation of motion.

Where

v = final velocity

u = Initial velocity

a = acceleration and

s = distance travelled

This equation gives us the velocity acquired by a body in traveling a distance s.

 

Points to remember:   

  1. If a body starts from rest, its initial velocity a = 0     
  2. If a body comes to rest (it stops), its final velocity, v = 0.
  3. If a body moves with uniform velocity, its acceleration, a = 0.