Question

Please help me. I am in 9th class please prove the 3 equations of motion.​

Answer 1

Deriving the three equations of motion...

Consider the acceleration of a body. The acceleration of a body is the rate of change in velocity of that body, from initial to final.

As time goes, the velocity of a body can be increased or decreased. The change in velocity measured in unit time is known as acceleration.

The velocity of a body at the starting point where we start to measure the velocity is known as its initial velocity (u), and that of the body at the ending point where we finish to measure the velocity is known as its final velocity (v).

According to these along with time (t), acceleration (a) of a body is expressed as,

$a=\dfrac{v-u}{t}$

Here we multiply 't' to both sides.

$at=v-u$

And now we add 'u' to both sides, and finally we get,

$\Large\boxed{v=u+at}$

Hence first equation of motion is derived!

Now consider the displacement (s) of a body. We know it is the product of the velocity of a body and time during which displacement occurs.

But here we're not sure about whether the body travelled in uniform velocity. Sometimes the velocity changes, means there occurs acceleration.

In this case, we take the average velocity of the body.

Here we take the median velocity as the average velocity.

I.e., we have initial velocity (u) and final velocity (v).

So the average velocity will be  $\dfrac{u+v}{2}.$

Hence displacement is expressed as,

$s=\left(\dfrac{u+v}{2}\right)t$

In RHS, according to the first equation of motion,

$s=\left(\dfrac{u+u+at}{2}\right)t\\ \\ \\ \Longrightarrow\ s=\left(\dfrac{2u+at}{2}\right)t\\ \\ \\ \Longrightarrow\ s=\left(u+\dfrac{at}{2}\right)t\\ \\ \\ \Longrightarrow\ s=\left(u+\dfrac{1}{2}at\right)t$

Now multiply the 't' into the brackets, and finally we get,

$\Large\boxed{s=ut+\dfrac{1}{2}at^2}$

Hence second equation of motion is derived!

Now consider the first equation of motion again.

$v=u+at$

We square both the sides of this equation.

$v^2=(u+at)^2\\ \\ \\ \Longrightarrow\ v^2=u^2+2uat+a^2t^2\\ \\ \\ \Longrightarrow\ v^2=u^2+2uat+\dfrac{2a^2t^2}{2}\\ \\ \\ \Longrightarrow\ v^2=u^2+2a\left(ut+\dfrac{at^2}{2}\right)\\ \\ \\ \Longrightarrow\ v^2=u^2+2a\left(ut+\dfrac{1}{2}at^2\right)$

In the bracket, we replace it by 's' according to second equation of motion. Thus we get,

$\Large\boxed{v^2=u^2+2as}$

Hence third equation of motion is derived!