Question

Derive the following formula for a body moving with uniform acceleration (a) V=u+at (b) S=u+1/2 at²​

Answer 1

$\large\bf\underline {To \: find:-}$

• we need to derive the first equation of motion and second equation of motion.

$\large\bf\underline{Given:-}$

• A body moving with uniform acceleration.

$\huge\bf\underline{Derivation:-}$

❥ First equation of motion :-

• v = u + at

we know that,

Acceleration = Change in Velocity/Time taken

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

$\dashrightarrow \: \: \: \: \tt \: at \: = v - u$

$\dashrightarrow \: \: \: \: \tt \:at - v = - u$

$\dashrightarrow \: \: \: \: \tt \: - v = - u - at$

$\dashrightarrow \: \: \: \: \tt \: - v = - (u + at)$

$\: \: \: \: \: \: \red{ \underline{ \boxed{ \tt \:v = u + at}}}$

Now,

❥ Second Equation of motion:-

• S = ut + ½at²

we know that,

Average velocity = (v + u)/2

• A = (v + u)/2

Distance travelled = Average velocity × time

$\dashrightarrow \: \: \: \: \tt \:s = ( \frac{v + u}{2} ) \times t$

putting v = u + at first equation of motion.

$\dashrightarrow \: \: \: \: \tt \:s = ( \frac{u + at + u}{2} ) \times t$

$\dashrightarrow \: \: \: \: \tt \:s = (\frac{2u + at}{2} ) \times t$

$\dashrightarrow \: \: \: \: \tt \: s = \frac{2ut}{2} + \frac{at {}^{2} }{2}$

$\dashrightarrow \: \: \: \: \tt \:s = ut + \frac{at {}^{2} }{2}$

$\red{ \underline{ \boxed{ \bf \: s = ut + \frac{1}{2}a {t}^{2} }}}$

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