Derive second equation of motion by calculus method.

To derive the second equation of motion using calculus, let's start with the equation for velocity:

v = u + at,

where:

v is the final velocity
u is the initial velocity
a is the acceleration
t is the time
Now, let's differentiate both sides of the equation with respect to time (t):

dv/dt = du/dt + a(d^2t/dt^2).

The derivative of velocity with respect to time (dv/dt) gives us the acceleration (a), and the derivative of initial velocity with respect to time (du/dt) gives us zero because the initial velocity is constant. Also, the derivative of time with respect to time (d^2t/dt^2) gives us the second derivative of time, which is the acceleration (a) since time is not changing its rate.

Therefore, the equation simplifies to:

a = a.

This equation is the second equation of motion, also known as the constant acceleration equation. It states that the acceleration of an object is constant over time.