Question

define third equation of motion by integration method

Answer 1

By definition, acceleration is the first derivative of velocity with respect to time. ... Unlike the first and second equations of motion, there is no obvious way to derive the third equation of motion (the one that relates velocity to position) using calculus.

Answer 2

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s = 1/2 t ( u + v )

Acceleration of the body ⬇

$\implies \: a = \frac{v - u}{t}$

From This,⬇️

$\implies \: t = \frac{v - u}{a}$

Substituting this in the above equation

$\implies \: = \frac{1}{2} ( \frac{v - u}{a} )(v + u) = \frac{1}{2} \frac{(v - u)(v + u)}{a} = \frac{(v {}^{2} { - u}^{2} }{2a}$

$\implies \: 2as = v {}^{2} - u {}^{2}$

$\implies \: v {}^{2} = u {}^{2} + 2as$

Note:-

This equation help's us to calculate the final velocity v of an object using u , a and s, even if the time taken to travel is unknown

$\implies \: v {}^{2} = u {}^{2} + 2as$

Consolidate the equations of motion :-

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$\implies \: v = u + at$

This is the first equation of motion

[ Velocity - Time relation ]

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$\implies \: s = ut + \frac{1}{2} at {}^{2}$

This is the second equation of motion

[ Position - Time relation ]

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$\implies \: v {}^{2} = u {}^{2} + 2as$

This is the Third equation of motion

[ Position - Velocity relation ]

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•This equations are applicable only to uniformly accelerated motion •.