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Updated: 13 Jun 2026, 12:22:16

E=mc²

First Equation of Motion

Given:

Initial velocity = u
Final velocity = v
Acceleration = a
Time = t

Formula:

v = u + at

Second Equation of Motion

Distance travelled in time t:

s = ut + (1/2)at²

Third Equation of Motion

Formula:

v² = u² + 2as

Derivation of the Third Equation

From the first equation:

v = u + at

Therefore,

t = (v - u)/a

Substitute into the second equation:

s = u[(v - u)/a] + (1/2)a[(v - u)/a]²

Simplifying:

s = (uv - u²)/a + (v - u)²/(2a)

Multiplying both sides by 2a:

2as = 2uv - 2u² + (v - u)²

Expanding:

2as = 2uv - 2u² + v² - 2uv + u²

2as = v² - u²

Therefore,

v² = u² + 2as

All Three Equations Together

v = u + at

s = ut + (1/2)at2

v2 = u2 + 2as

These are commonly called Newton's equations (or equations of uniformly accelerated motion). The powers are written in the form you requested, such as v², u², and t².

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Swarnim Raj

Author & Career Content Writer

Experienced education and career writer at Infokept Career Hub, creating simple, research-based guides for students and government job aspirants.

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