A particle moves on a circle of radius `r` with centripetal acceleration as function of time as `a_c = k^2 r t^2`, where `k` is a positive constant. Find the following quantities as function of time at an instant :
(a) The speed of the particle
(b) The tangential acceleration of the particle
( c) The resultant acceleration, and
(d) Angle made by the resultant acceleration with tangential acceleration direction.

Explanation

To solve the problem step by step, we will find the required quantities based on the given centripetal acceleration function \( a_c = k^2 r t^2 \).

### Step 1: Find the Speed of the Particle

The centripetal acceleration \( a_c \) is related to the speed \( v \) of the particle moving in a circle of radius \( r \) by the formula:

\[
a_c = \frac{v^2}{r}
...