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 consider the time it takes for each ball to reach that height.

Let's break down the problem step by step:

1. The first ball is launched straight up with initial speed v0. We can use the equation for the vertical position of an object in free-fall motion:

y = v0*t - (1/2)*g*t^2

where y is the vertical position, v0 is the initial velocity, g is the acceleration due to gravity, and t is the time.

At the top of the trajectory, the vertical position is the maximum height reached by the first ball. We'll denote it as h1.

So, at the top of the trajectory, y1 = h1 = v0*t1 - (1/2)*g*t1^2, where t1 is the time taken by the first ball to reach the maximum height.

2. The second ball is also launched with the same initial speed v0 from the top of its trajectory. Let's denote the time taken by the second ball to reach the maximum height as t2.

Since the initial velocity of the second ball is zero, its equation becomes: y = - (1/2)*g*t^2

At the top of the trajectory, y2 = h1 (the same maximum height reached by the first ball).

3. Since the two balls collide at the same vertical position at the top of the trajectory, y1 = y2. Therefore, we can equate the two equations:

v0*t1 - (1/2)*g*t1^2 = - (1/2)*g*t2^2

4. We can rearrange the equation to find the ratio t1/t2:

v0*t1 - (1/2)*g*t1^2 = - (1/2)*g*t2^2
v0*t1 = (1/2)*g*(t2^2 - t1^2)
v0*t1 = (1/2)*g*(t2 + t1)*(t2 - t1)
t1/t2 = (v0*(t2 + t1)) / (2*g*(t2 - t1))

Now, you can plug in the values for the initial speed v0 and the acceleration due to gravity g to calculate the ratio t1/t2. This ratio represents the fraction of the maximum height reached by the first ball at which the two balls collide.