Question

What is integral of the equation of motion​

Answer 1

Answer:

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

⇒ v₀∫v 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

⇒ s₀∫s ds = ₀∫t (v₀ + at) dt

⇒ s−s0=[v0t+at22]t0,

⇒ 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

⇒ v₀∫v v dv = s₀∫s a x ds

⇒ [v22]vv0=a[s]ss0,

⇒ ½ [v² – v²₀] = a [s – s₀]

⇒ v² – v²₀ = 2a [s – s₀]

∴ v² = v²₀ + 2a [s – s₀].