Question

Derive the equation of motion by the calculus or graphical method?

Answer 1

By using calculus

1). Acceleration is given as rate of change of velocity

$a=\frac{dv}{dt}$

Now on integrating both sides

$a{\int }_0^{t}dt={\int }_u^{v}dv$

On solving the above we get

$v=u+at$

2). Rate of the displacement is equal to velocity

$v=\frac{ds}{dt}$

On integrating the above equation we get

${\int }_0^{s}ds=v{\int }_0^{t}dt$

On substituting the value of$v=u+at$

We get

$s=ut+\frac{1}{2}a{t}^2$

3). Distance travelled is equal to average speed time time

$s=\frac{u+v}{2}t$

By using the first equation and substituting the value of$t=\frac{v-u}{a}$

We get

$v^2=u^2+2as$