Paige and Julia are playing volleyball. Paige is in spiking position when Julia gives her the perfect set. The 0.226-kg volleyball is 2.29 m above the ground and has a speed of 1.06 m/s. Paige spikes the ball, doing 9.89 J of work on it. Determine a) the total mechanical energy of the ball after the spike, and b) the speed of the ball upon hitting the ground on the opponent's side.
To determine the total mechanical energy of the ball after the spike, we need to consider both its kinetic energy and potential energy.
a) The initial potential energy of the ball is given by PE = mgh, where m is the mass of the ball (0.226 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the ground (2.29 m).
PE = (0.226 kg) * (9.8 m/s^2) * (2.29 m) = 5.22 J
The initial kinetic energy of the ball is given by KE = (1/2)mv^2, where m is the mass of the ball (0.226 kg) and v is the speed of the ball (1.06 m/s).
KE = (1/2) * (0.226 kg) * (1.06 m/s)^2 = 0.127 J
The total mechanical energy after the spike is the sum of the potential and kinetic energies:
Total mechanical energy = PE + KE = 5.22 J + 0.127 J = 5.35 J
b) To determine the speed of the ball upon hitting the ground on the opponent's side, we need to find the potential energy at that point.
Since the ball is now at ground level, the potential energy is zero. The kinetic energy at this point is equal to the total mechanical energy before the spike, which is 5.35 J.
KE = (1/2)mv^2
Rearranging the equation to solve for v:
v = sqrt((2 * KE) / m)
v = sqrt((2 * 5.35 J) / 0.226 kg)
v ≈ sqrt(47.3) ≈ 6.88 m/s
Therefore, the speed of the ball upon hitting the ground on the opponent's side is approximately 6.88 m/s.
To find the total mechanical energy of the ball after the spike, we need to understand the concept of mechanical energy. Mechanical energy is the sum of kinetic energy and potential energy. In this case, the potential energy is due to the height of the volleyball above the ground, and the kinetic energy is due to its speed.
a) To find the potential energy of the volleyball, we can use the formula:
Potential Energy = mass * acceleration due to gravity * height
Since the mass of the volleyball is 0.226 kg, the acceleration due to gravity is approximately 9.8 m/s^2, and the height is 2.29 m, we can calculate:
Potential Energy = 0.226 kg * 9.8 m/s^2 * 2.29 m = 4.953 J
Now, let's determine the kinetic energy of the volleyball using the formula:
Kinetic Energy = 1/2 * mass * velocity^2
With the mass of the volleyball as 0.226 kg and the velocity as 1.06 m/s, we can calculate:
Kinetic Energy = 1/2 * 0.226 kg * (1.06 m/s)^2 = 0.125 J
The total mechanical energy is the sum of the potential energy and kinetic energy:
Total Mechanical Energy = Potential Energy + Kinetic Energy = 4.953 J + 0.125 J = 5.078 J
Therefore, the total mechanical energy of the ball after the spike is 5.078 J.
b) To determine the speed of the ball upon hitting the ground on the opponent's side, we need to apply the principle of conservation of energy. According to this principle, the total mechanical energy before the spike should equal the total mechanical energy after the spike.
Since the ball was in spiking position, its initial potential energy was zero. Therefore, the initial total mechanical energy is only the kinetic energy:
Initial Total Mechanical Energy = Kinetic Energy = 0.125 J
Since the total mechanical energy after the spike is 5.078 J, we can set up the equation:
Initial Total Mechanical Energy = Final Total Mechanical Energy
0.125 J = 5.078 J
Now, to solve for the final kinetic energy, we subtract the initial kinetic energy from the total mechanical energy after the spike:
Final Kinetic Energy = Final Total Mechanical Energy - Initial Kinetic Energy
Final Kinetic Energy = 5.078 J - 0.125 J = 4.953 J
Finally, we can use the formula for kinetic energy to find the final velocity of the ball:
Final Kinetic Energy = 1/2 * mass * final velocity^2
Rearranging the formula:
final velocity^2 = (2 * Final Kinetic Energy) / mass
Plugging in the values of final kinetic energy as 4.953 J and mass as 0.226 kg:
final velocity^2 = (2 * 4.953 J) / 0.226 kg = 21.88
Taking the square root of both sides:
final velocity = sqrt(21.88) = 4.67 m/s (approximately)
Therefore, the speed of the ball upon hitting the ground on the opponent's side is approximately 4.67 m/s.