In uniform circular motion, the acceleration of a body is directed towards the center of the circle and is called centripetal acceleration. The magnitude of the centripetal acceleration is given by the equation:
a_c = (v^2) / r
Where: a_c is the centripetal acceleration, v is the velocity of the body, and r is the radius of the circular path.
If the initial velocity is doubled while the radius remains unchanged, we can compare the initial and final velocities. Let's denote the initial velocity as v_i and the final velocity as v_f.
v_f = 2 * v_i
Now, let's calculate the ratio of the final velocity to the initial velocity:
v_f / v_i = (2 * v_i) / v_i = 2
So, the final velocity is twice the initial velocity.
Since the radius remains unchanged, the centripetal acceleration (a_c) is only dependent on the velocity (v) and the radius (r). Therefore, the centripetal acceleration remains constant throughout the motion.
Hence, the average acceleration of the body in uniform circular motion, when the initial velocity is doubled but the radius remains unchanged, is equal to the centripetal acceleration, which is a_c = (v^2) / r.