A basketball is thrown with a velocity of 10.5 ft/s downward from a height of 6.8 ft. The coefficient of restitution between the ball and the floor is 0.65. How high does the basketball go after its 2nd bounce?

To find out how high the basketball goes after its 2nd bounce, we need to use the concept of conservation of mechanical energy.

First, let's calculate the initial potential energy of the basketball before the first bounce. The potential energy (PE) is given by the formula PE = mgh, where m is the mass of the basketball, g is the acceleration due to gravity, and h is the initial height.

Next, we can calculate the initial kinetic energy of the basketball before the first bounce. The kinetic energy (KE) is given by the formula KE = (1/2)mv^2, where m is the mass of the basketball and v is the initial velocity.

Since no energy is lost during the bounce (assuming a perfect ball and floor collision), the total mechanical energy (E) of the basketball remains constant. Therefore, the sum of the initial potential energy and initial kinetic energy is equal to the sum of the final potential energy and final kinetic energy.

E_initial = PE_initial + KE_initial
E_final = PE_final + KE_final

Since the basketball reaches its maximum height after the first bounce, the final kinetic energy is zero. Therefore, E_final = PE_final.

Now, let's go through the calculations step by step:

Step 1: Calculate the initial potential energy (PE_initial):
PE_initial = mgh
= mass × gravity × initial height
= m × 9.8 m/s^2 × 6.8 ft

Step 2: Calculate the initial kinetic energy (KE_initial):
KE_initial = (1/2)mv^2
= (1/2) × mass × (velocity)^2
= (1/2) × m × (10.5 ft/s)^2

Step 3: Calculate the total mechanical energy (E):
E_initial = PE_initial + KE_initial

Step 4: Calculate the final potential energy (PE_final):
PE_final = E_final
= E_initial

Now, we have calculated the energy at each stage. To find the height of the basketball after the 2nd bounce, we can use the equation PE_final = mgh, and solve for h.

PE_final = mgh
E_initial = mgh
PE_final = E_initial

Solving for h, we have:

h = E_final / (mg)

Plug in the values of E_final (which is equal to E_initial), mass (if given), and the acceleration due to gravity (which is approximately 9.8 m/s^2) to find the height.

Note: The coefficient of restitution (COR) is not necessary to calculate the height in this scenario. It is used to determine the velocity after a collision.