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B31. Great basketball players sometimes seem to hang in the air while they do l...

Sep 25, 2024

Solution

To find the time in the air for a vertical jump of 1.2 m, we can use the kinematic equation for free fall. The total time of flight for a jump is twice the time taken to reach the maximum height. The equation we will use is:

d = v_i t + \frac{1}{2} a t^2

. At the peak of the jump, the final velocity (

v_f

) is 0, and the initial velocity (

v_i

) can be found using the equation

v_f^2 = v_i^2 + 2ad

. Here,

d = 1.2 m

and

a = -9.8 m/s^2

. Rearranging gives us:

0 = v_i^2 - 2(9.8)(1.2)

, which leads to

v_i = \sqrt{2(9.8)(1.2)} \approx 4.85 m/s

. Now, we can find the time to reach the peak:

t = \frac{v_i}{g} = \frac{4.85}{9.8} \approx 0.495 s

. Therefore, the total time in the air is approximately

2t \approx 0.99 s

To find the height a person would have to leap to have a "hang time" of 2.0 s, we first find the time to reach the peak, which is half of the total hang time:

t = \frac{2.0 s}{2} = 1.0 s

. Using the equation

d = v_i t + \frac{1}{2} a t^2

, we need to find the initial velocity (

v_i

) at the peak. We can use

v_i = g t = 9.8 \times 1.0 = 9.8 m/s

. Now, substituting into the displacement equation gives us:

d = (9.8)(1.0) + \frac{1}{2}(-9.8)(1.0^2) = 9.8 - 4.9 = 4.9 m

. Thus, the height required for a hang time of 2.0 s is approximately 4.9 m

Answer

a: 0.99 s

b: 4.9 m

Key Concept

Kinematics of Free Fall: The motion of an object under the influence of gravity can be described using kinematic equations. The acceleration due to gravity is approximately

9.8 m/s^2

, and the time of flight can be calculated using the height of the jump. The total time in the air is twice the time to reach the maximum height.

Explanation

The calculations for both questions utilize the principles of kinematics, specifically the equations of motion under constant acceleration due to gravity, to determine the time in the air and the height needed for a specified hang time.