Question

5. Derive the following equations for uniformly accelerated motion from velocity-time graph.
(i) v = u + at (ii) s = ut + ½at2(iii) v2-u2 =2as
where symbols have their usual
meanings.​

Answer 1

Graphical Derivation:

• Let's assume that Velocity-Time graph has been provided.

• The form of the graph is y = mx + c.

• Similarly, we can say:

$\boxed{ \sf \: v = u + at}$

• Here 'a' is the slope of the graph = acceleration.

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Now, displacement is given by area under v-t graph.

$\sf \: d = \dfrac{1}{2} \times (u + v) \times t$

$\sf \implies \: d = \dfrac{1}{2} \times (u + u + at) \times t$

$\sf \implies \: d = \dfrac{1}{2} \times (2u+ at) \times t$

$\sf \implies \: d = ut + \dfrac{1}{2} a {t}^{2}$

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$\sf \: s = \dfrac{u + v}{2} \times t$

$\sf \implies\: s = \dfrac{u + v}{2} \times \dfrac{v - u}{a}$

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

[Hence Proved]

Hope It Helps.