I fire a tennis ball from an air cannon, straight up from ground level, at initial speed v0. At

the instant it reaches the top of its trajectory, I fire a second tennis ball at the same initial
speed. Air resistance is negligible. At what fraction of the maximum height reached by the
first ball do the two collide?

To find the fraction of the maximum height reached by the first ball at which the two balls collide, we need to analyze the motion of the tennis balls.

Let's break down the problem step-by-step:

1. Vertical motion of the first ball:
When the first ball is fired straight up, it experiences a free-fall motion due to the force of gravity. The initial speed v0 determines how high the ball will go before gravity brings it back down. The time it takes for the ball to reach its maximum height is given by the formula:

t_max = v0 / g

where g is the acceleration due to gravity.

2. Maximum height of the first ball:
To find the maximum height H_max reached by the first ball, we can use the kinematic equation for vertical motion:

H_max = (v0^2) / (2g)

3. Time of flight of the first ball:
The total time of flight, T, for the first ball can be calculated as twice the time it takes to reach its maximum height:

T = 2 * t_max = 2 * (v0 / g)

4. Vertical motion of the second ball:
Since the second ball is fired at the instant the first ball reaches its maximum height, it will start moving from the same height H_max. The time it takes for the second ball to reach this height is the same as the time of flight T for the first ball.

5. Colliding at the moment:
Since the second ball takes the same time T to reach H_max as the first ball takes to reach its maximum height and start descending, the two balls will collide at that moment, forming a fraction of the maximum height H_max.

Thus, the fraction of the maximum height at which the two balls collide is 1.