The NCAA requires that a basketball dropped from a height of 1.83 m (measured to the bottom
of the ball) bounce to a height of 1.24 to 1.37 m (measured to the top of the ball). Assuming the
diameter of a typical basketball is 0.25 m, calculate the allowable range of coefficients of
restitution between the basketball and the gym floor.
To calculate the allowable range of coefficients of restitution between the basketball and the gym floor, we need to understand the concept of coefficients of restitution and use the given information.
The coefficient of restitution (COR) is a measure of how "bouncy" an object is when it collides with another object. It is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of approach before the collision.
In this case, we can use the following formula to calculate the COR:
COR = (v₂ - v₁) / (u₁ - u₂)
Where:
- v₁ is the velocity of the basketball just before it hits the gym floor.
- v₂ is the velocity of the basketball just after it bounces off the gym floor.
- u₁ is the initial velocity of the basketball just before it hits the gym floor.
- u₂ is the initial velocity of the floor, which is 0 since it is stationary.
Since the basketball is dropped from a height of 1.83 m, we can use the concept of conservation of energy to find the initial velocity u₁. The potential energy at the top should be equal to the kinetic energy at the bottom.
Potential energy at the top = Kinetic energy at the bottom
mgh = 0.5mu₁²
Where:
- m is the mass of the basketball (which we can assume to be constant for all basketballs).
- g is the acceleration due to gravity (approximately 9.8 m/s²).
- h is the height from which the ball is dropped (1.83 m).
- u₁ is the initial velocity of the basketball just before it hits the gym floor.
We can rearrange this equation to solve for u₁:
u₁ = sqrt(2gh)
Now, we can substitute the given values and calculate u₁:
u₁ = sqrt(2 * 9.8 * 1.83) ≈ 7.57 m/s
Next, we need to determine the velocity of the basketball just after it bounces off the gym floor (v₂). This can be calculated using the COR and the equation:
v₂ = COR * (u₁ - u₂) + u₂
Since u₂ is 0 (as the floor is stationary), the equation simplifies to:
v₂ = COR * u₁
We can substitute the given bounce height range of 1.24 to 1.37 m to find the allowable range of COR:
1.24 ≤ COR * u₁ ≤ 1.37
Now, we can rearrange the inequality to solve for the range of COR:
1.24 / u₁ ≤ COR ≤ 1.37 / u₁
Finally, we substitute the value of u₁ ≈ 7.57 m/s to calculate the allowable range of COR:
1.24 / 7.57 ≤ COR ≤ 1.37 / 7.57
0.1638 ≤ COR ≤ 0.1811
Therefore, the allowable range of coefficients of restitution between the basketball and the gym floor is approximately 0.1638 to 0.1811.