Question

Can you derivative the third equation of motion without graphical method?​

Answer 1

Answer:

The third equation of motion:

$\boxed{ \red{\sf{ {v}^{2} = {u}^{2} + 2a S }}}$

Derivation without graphing:

Distance travelled by an object = avg. velocity × time

$\sf{avg. \: velocity = \frac{u + v}{2} }$

Where u and v are the initial and final velocities respectively.

Let time be denoted by t:

$\sf{displacement = (\frac{u + v}{2}) \times t \: ... _{(i)} }$

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From the first equation of motion:

$\boxed{ \blue{\sf{ v = u + a t }}}$

Solving for t:

$\implies\sf{ v - u = a t }$

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

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In eqn (I), substituting the given value for t:

$\implies \sf{displacement = (\frac{u + v}{2}) \times (\frac{v - u}{a} ) \: }$

(u + v)(v - u) on expanding make v² - u²

$\implies \sf{displacement = \frac{ {v}^{2} - {u}^{2} }{2a} }$

Using S for displacement:

$\implies \sf{S = \frac{ {v}^{2} - {u}^{2} }{2a} }$

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

Rearranging the terms:

$\boxed{ \green{ \implies \sf{ {v}^{2} = {u}^{2} + 2a S}}}$

There! We got the third equation of motion without mentioning graphs.

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Arigato! Hope this helps! :)