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].