Answer to Question #2797 in Mechanics | Relativity for muskan

Question

Prove graphically and anatically following equations of motions;
1.) v = u+at
2.)v[sup]2[/sup] = u[sup]2[/sup]+2as
3.)s = ut+1/2at[sup]2[/sup]

Answer 1

Let x(t) be the position of some body. Let v(t) be the velocity of some body. Than its position at any moment of the time will be
∫v₀ dt = v₀ t + x₀ = x(t)
where x₀ is the initial position.
If body has nonzero acceleration than
at²/2+v₀ t+x₀=x(t)
where a is the acceleration.
If x₀=0 we get formula 3.
at²/2 + v₀ t = x(t) - x₀ = x(t) = S.
Differentiating this expression we get
x' (t) = v₀+at = v(t)
And that is formula 1.
v = v₀+at
S = v₀ t+(at²)/2
t = (v-v₀)/a
S=v₀ (v-v₀)/a+(a(v²-2v² v₀² + v₀²))/(2a² ) = (v²-v₀²)/2a

a=(v²-v₀²)/2S
v²=v₀²+2Sa.

This is formula 2.

This graph illustrate function
v₀t+x₀=x(t).

This graph illustrates function
at²/2 + v₀t + x₀ = x(t)
Area under the line for given interval of t gives S.