<p>Two basketball players are essentially equal in all respects. (They are the s

Question

<p>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&#160;<span>(his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.</span></p>
<p>Find the vertical displacement D(t)=h_A(t)-h_B(t),&#160;as a function of time for the interval 0&lt;t&lt;t_R, where h_A(t)&#160;is the height of the raised hands of Arabella, while h_B(t)&#160;<span>is the height of the raised hands of Boris.&#160;<span>Express the vertical displacement in terms of H, g, and t.</span></span></p>

Explanation / Answer

If their initial velocity is v0, then the height at any time t is h0 + V0*t - 0.5*g*t² Since both can jump to height H, the initial velocity is given by m*g*H = 0.5*m*v0². Thus v0 =v[2*g*H] Thus ha = h0 + v[2*g*H]*t - 0.5*g*t² and hb = h0 + v[2*g*H](t - tr) - 0.5*g*(t - tr)² If t tr The difference ?h = ha - hb =h0 + v[2*g*H]*t - 0.5*g*t² - h0 - v[2*g*H](t - tr) + 0.5*g*(t - tr)² ?h = v[2*g*H]*tr - g*tr*t + 0.5*g*tr² To minimize the chance of the shot being blocked, ?h should be max. This occurs at t = tr, the instant Boris leaves the ground. ?h starts at 0 and increases according to v[2*g*H]*t - 0.5*g*t² until t = tr and ?h(tr) = v[2*g*H]*tr - 0.5*g*tr². After that ?h = v[2*g*H]*tr - g*tr*t + 0.5*g*tr² which is a decreasing function of t starting at ?h(tr).

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