Question

Give the mathematical (calculus method) method for derivation of equations of motion.

Answer 1

Suppose body is moving in straight line with uniform acceleration a. Its initial velocity (i.e. at t = 0) is u and after time t is u. Total distance travelled by the body in this time is S.

First equation: According to definition of instantaneous acceleration

a = \(\frac{dv}{dt}\)

or dv = adt

On integrating both sides,

Second equation: According to definition of instantaneous velocity,

v = \(\frac{dS}{dt}\) 

or dS = v.dt = (u + at).dt

On integrating both sides,

Third equation: According to definitions of instantaneous velocity and acceleration,

We know that the instantaneous acceleration

a = \(\frac{dv}{dt}\) ............(1)

We know that the instantaneous velocity

v = \(\frac{ds}{dt}\) ...........(2)

From equation (1) and (2), we have

∴ \(\frac{a}{v} = \frac{dv/dt}{ds/dt} = \frac{dv}{ds}\)

∵ a ds = v dv ...........(3)

On integrating both sides,

Displacement of body in a particular second:

Suppose an object is moving under uniform acceleration a and we have to find the distance travelled in t^th second.

∵ Distance travelled in t second is obtained by:

S = ut + \(\frac{1}{2} \) at²

∴ Distance travelled in t^th second