A basketball star covers 2.70 m horizontally in a jump to dunk the ball (see figure). His motion through space can be modeled as that of a particle at a point called his center of mass (which we shall define in Chapter 9). 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 0.940 m when he touches down again.
(a) Determine his time of flight (his "hang time").
To determine the time of flight or hang time of the basketball star, we can use the equation for the vertical motion of a particle:
Δy = V₀y * t + (1/2) * g * t²
where:
Δy is the change in height
V₀y is the initial vertical velocity
t is the time of flight
g is the acceleration due to gravity
We are given the following information:
- The initial height, y₀, is 1.02 m
- The maximum height, y_max, is 1.80 m
- The final height, y_final, is 0.940 m
Since the basketball star starts and ends at the same height, the change in height is zero:
Δy = y_final - y₀ = 0.940 m - 1.02 m = -0.080 m
The initial vertical velocity, V₀y, can be calculated using the equation:
V₀y² = V₀² - 2 * g * Δy,
where V₀ is the initial velocity of the basketball star. Since we are given the initial velocity along the horizontal direction, we assume that there is no vertical velocity initially:
V₀y = 0 m/s
Substituting the values into the equation, we get:
0 = 0 - 2 * g * Δy
2 * g * Δy = 0
Since the equation is equal to zero, we can say that the value of Δy is equal to zero. This means that the basketball star doesn't change his vertical position during the jump. Therefore, the time of flight is determined by the equation:
t = 2 * t_peak,
where t_peak is the time taken to reach the maximum height. We can calculate t_peak using the equation:
V_fy = V₀y - g * t_peak
Since the vertical velocity at the maximum height is zero, we can write:
V_fy = 0 m/s
0 = 0 - g * t_peak
Simplifying the equation, we find:
t_peak = 0 s
Therefore, the time of flight or hang time of the basketball star is:
t = 2 * t_peak = 2 * 0 s = 0 s
The hang time is zero. This might seem counterintuitive, but it can happen in certain idealized situations where there is no air resistance. In reality, there would be some hang time, but it would be very small.