Question

A basketball star covers 2.70 m horizontally in a jump to dunk the ball (see figure). His motion through space can be modeled precisely as that of a particle at his center of mass. His center of mass is at elevation 1.02 m when he leaves the floor. It reaches a maximum height of 1.80 m above the floor and is at elevation 0.950 m when he touches down again.

(a) Determine his time of flight (his "hang time").
(b) Determine his horizontal velocity at the instant of takeoff.
(c) Determine his vertical velocity at the instant of takeoff.
(d) Determine his takeoff angle.above the horizontal
(e) For comparison, determine the hang time of a whitetail deer making a jump (see figure above) with center-of-mass elevations yi = 1.20 m, ymax = 2.50 m, and yf = 0.720 m.

Answer 1

Part a)

`T = 0.81 s`

Part b)

`v_x = 3.33 m/s`

Part c)

`v_y = 3.91 m/s`

Part d)

`\theta = 49.55 degree`

Part e)

`T = 1.11 s`

Step-by-step explanation:

Part a)

initial vertical position = 1.02 m

maximum height = 1.80 m

`\Delta y = 1.80 - 1.02`

`\Delta y = 0.78 m`

`v_f^2 - v_y^2 = 2a \Delta y`

`0 - v_y^2 = 2(-9.81)(0.78)`

`v_y = 3.91 m/s`

time taken by it to reach this height

`v_y = v_i + at`

`0 = 3.91 - 9.81 t_1`

`t_1 = 0.39 s`

Now when it again touch the ground then its speed is given as

`v_f^2 - v_y^2 = 2a \Delta y`

`v_f^2 - 0 = 2(9.81)(1.80 - 0.95)`

`v_y = 4.08 m/s`

time taken by it to reach this height

`4.08 = v_i + at`

`4.08 = 0 + 9.81 t_2`

`t_2 = 0.42 s`

`T = t_1 + t_2`

`T = 0.81 s`

Part b)

Horizontal velocity

`v_x = (x)/(t)`

`v_x = (2.70)/(0.81)`

`v_x = 3.33 m/s`

Part c)

vertical velocity is the intial y direction velocity

`v_y = 3.91 m/s`

Part d)

Take off angle is given as

`tan\theta = (3.91)/(3.33)`

`\theta = 49.55 degree`

Part e)

initial vertical position = 1.20 m

maximum height = 2.50 m

`\Delta y = 2.50 - 1.20`

`\Delta y = 1.30 m`

`v_f^2 - v_y^2 = 2a \Delta y`

`0 - v_y^2 = 2(-9.81)(1.30)`

`v_y = 5.05 m/s`

time taken by it to reach this height

`v_y = v_i + at`

`0 = 5.05 - 9.81 t_1`

`t_1 = 0.51 s`

Now when it again touch the ground then its speed is given as

`v_f^2 - v_y^2 = 2a \Delta y`

`v_f^2 - 0 = 2(9.81)(2.50 - 0.72)`

`v_y = 5.9 m/s`

time taken by it to reach this height

`5.9 = v_i + at`

`5.9 = 0 + 9.81 t_2`

`t_2 = 0.60 s`

`T = t_1 + t_2`

`T = 1.11 s`