# Equation of Motion by Integration Method

## Equation of Motion by Integration Method

Relation among velocity, distance, time and acceleration is called equations of motion. There are three equations of motion.

The final velocity (v) of a moving object with uniform acceleration (a) after time (t).

Let, Initial velocity = v₀,

Final velocity = v,

Time = t,

Acceleration = a

First Equation of Motion: Acceleration is the first derivative of velocity with respect to time.

Acceleration (a) = dv/dt

⇒ dv = a x dt

v₀v dv = ₀∫t a dt

⇒ (v – v₀) = a (t – 0)

⇒ v – v₀ = at

∴ v = v₀ + at

Second Equation of Motion: Velocity is the first derivative of position with respect to time.

⇒ Velocity (v) = ds/ dt

⇒ ds = v dt

⇒ ds = (v₀ + at) dt

s₀s ds = ₀∫t (v₀ + at) dt

⇒ $$s-{{s}_{0}}=\left[ {{v}_{0}}t+\frac{a{{t}^{2}}}{2} \right]_{0}^{t}$$,

⇒ s – s₀ = v₀t + (at²/2)

∴ s = s₀ + v₀t + (at²/2)

Third Equation of Motion:

Acceleration (a) = dv/ dt

= (dv/ ds) x (ds/ dt)

= (dv/ ds) x v

∴ Acceleration (a) = v dv/ ds

⇒ v dv/ ds = a

v₀v v dv = s₀s a x ds

⇒ $$\left[ \frac{{{v}^{2}}}{2} \right]_{{{v}_{0}}}^{v}=a\left[ s \right]_{{{s}_{0}}}^{s}$$,

⇒ ½ [v² – v²₀] = a [s – s₀]

⇒ v² – v²₀ = 2a [s – s₀]

∴ v² = v²₀ + 2a [s – s₀].