Question

Derive s=ut+1/2at^2 by calculus method

Answer 1

Answer:

Explanation:

The given equation is the second equation of motion and is derived through the calculus method as follows –

As we know that the velocity is the measure of displacement over time. It can also be said as the rate of displacement

Thereby, velocity = rate of displacement

v = d s/ dt

Small displacement over small time

ds = v dt

From first equation of motion, we have –

v = u + at

d s = v dt

ds = ( u + at ) dt

Integrate the above equation –

$\begin{array}{l}{\int_{0}^{s} d s=\int_{0}^{t} u d t+\int_{0}^{t} a t d t} \\ {s=u t+\frac{1}{2} a t^{2}}\end{array}$

This is how the second equation of motion is derived.

Answer 2

Answer:

Explanation:

s=ut+12at2  is one of the equations of motion, where t = time, u = initial velocity, a = acceleration and s = displacement.

Taking the first derivative with respect to time:

dsdt=d(ut+0.5at2)dt

dsdt is the rate of change of displacement with respect to time, which is known as velocity.

v=u+at

We now have an equation that links the velocity at a given time to the initial velocity and the acceleration.

Taking the second derivative:

d2sdt2=dvdt=d(u+at)dt=a

d2sdt2 is the rate of change of velocity with respect to time, which is known as acceleration.