Question

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, (called H 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 , and Boris jumps later, at time (his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.

1)Find the vertical displacement , D(t) = H(a)(t) - H(b)(t) as a function of time for the interval 0 < t < t(r) , where H(a)(t) is the height of the raised hands of Arabella, while H(b)(t) is the height of the raised hands of Boris.

Express the vertical displacement in terms of H, g , and t.

Answer 1

The vertical displacement D(t) between two jumping basketball players can be found using the kinematic equation for vertical motion. By accounting for the later jump time of the second player, we can subtract their height functions to find the displacement as a function of time in terms of initial height H, gravitational acceleration g, and time t.

Step-by-step explanation:

The student's question involves calculating the vertical displacement of two basketball players, Arabella and Boris, during their jumps, with Arabella starting her jump before Boris. To find the vertical displacement D(t) as a function of time, we need to consider the kinematic equation for vertical motion H(t) = V0t - (1/2)gt2, where V0 is the initial vertical velocity, g is the acceleration due to gravity (9.8 m/s2), and t is the time.

Assuming both players have the same V0 and are affected by the same g, the displacement of each player at any time t is given by:

• For Arabella: H(a)(t) = V0t - (1/2)gt2
• For Boris: H(b)(t) = V0(t - tr) - (1/2)g(t - tr)2

Given that Boris starts jumping at time tr later than Arabella, his height function H(b) has t - tr. The vertical displacement D(t) between Arabella and Boris from 0 < t < tr can be found by subtracting the height of Boris from the height of Arabella:

D(t) = H(a)(t) - H(b)(t)

Substituting in the expressions for H(a)(t) and H(b)(t) gives us the final expression for D(t) in terms of the known quantities H, g, and t.

Answer 2

The reaction time of Boris is t(r), so before that, Boris will not have jumped. Thus, H(b)(t) = 0
The vertical displacement will simply be
D(t) = H(a)(t)