Question

Describe the three equations for uniform accelerated motion.

Answer 1

Calculus method :

(i) Velocity-time relation : These equations can also be derived from calculus method. From the definition of acceleration;
a = \(\frac{d v}{d t}\) or dv = a dt
Integrating it with in the condition of motion (i.e.) when time changes from 0 to t, velocity changes from u to v, we get

This is the first equation of motion.

(ii) Distance time relation: The instantaneous velocity of an object in uniformly accelerated motion is given by
v = \(\frac{d x}{d t}\) or dx = v dt
v = u + at
∴ dx = (u + at) dt ………….. (vii)
Let the displacement of the object from the origin of position-axis is x₀ at t = 0 and x at t = t, integrating both the sides of the equation (vii) within proper limits, we have,

If x - x₀ = S = distance covered by the object in time t, then from eq. (viii)
s = ut + 1/2 at² …………… (ix)
This is the second equation of uniform accelerated motion.

(iii) Velocity-displacement relation
The instantaneous acceleration is given by

Let u and v be the velocity of the object at positions given by displacements x₀ and x.
Integrating the above equation (x) with the condition of motion, we get

This is the third equation of uniform acceleration motion.