Two basketball players are essentially equal in all respects. (They are the same height, they jump with the same initial velocity, etc.) In particular, by jumping they can raise their centers of mass the same vertical distance, H (called their "vertical leap"). The first player, Arabella, wishes to shoot over the second player, Boris, and for this she needs to be as high above Boris as possible. Arabella jumps at time t=0, and Boris jumps later, at time t_R (his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.

Calculate a formula for their jump heights depending on the time and look at the derivative of the difference to see, when Arabella's height is maximal in relation to Boris.

There's actually no question posted here, just a series of statements. I assume you want to know, "when should Arabella release the ball"?

Arabella's height is given by:\[s_{A}(t)=x _{0}+v_{0}t-\frac{1}{2} g t^2\]Where x0 is the height of her center of mass, v0=initial velocity of her jump, g is gravity and t is time. Boris', position function is:\[s_{B}(t)=x_{0}+v_{0}(t-R)-\frac{1}{2} g (t-R)^2\] Where R is his reaction time. We want to maximize the difference in Arabella's and Boris' heights. That is we want to find the max of:\[s_{A}(t)-s_{B}(t)=(x _{0}+v_{0}t-\frac{1}{2} g t^2)-(x_{0}+v_{0}(t-R)-\frac{1}{2} g (t-R)^2)\]Simplifying:\[=v_{0}R-\frac{1}{2} g t^2-\frac{1}{2}g(t^2-2tR+R^2)=v_{0}R- g t^2+gtR-\frac{1}{2}gR^2\]Now we need to use calculus to maximize this function.

Let's find the critical values of this function by taking it's derivative (keeping in mind that v0, R and g are all constants) and setting it equal to zero:\[0=gR-2 g t\]This gives:\[t=\frac{1}{2}R\]We note that this critical value occurs at a maximum (one way to see this is using the second derivative test which is equal to -2g; thus the function is concave downward). Arabella should release the ball at time t=(1/2)R

Actually, this answer is wrong. The best Arabella can do is shoot the ball at t=R. If you're curious why, ask.

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