You drop a ball from a height of 2.0 m and it bounces back to a height of 1.5 m (a) What fraction of its initial energy is lost during the bounce? (b) What is the ball’s speed just at it leaves the ground after the bounce?
(a) 1/4
(b) sqrt(3/4)*sqrt(2gH)
where H = 2.0 m
You could also write it as sqrt(2gH'), where H' = 1.5 m is the distance it rises after the first bounce
To find the fraction of the ball's initial energy that is lost during the bounce, we need to compare the potential energy before and after the bounce. We can use the law of conservation of energy, which states that energy is neither created nor destroyed, only transferred from one form to another. Here's how to calculate it:
(a) Fraction of initial energy lost:
1. Determine the potential energy before the bounce: The potential energy (PE) of an object near the Earth's surface is given by the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
Since the ball is dropped from a height of 2.0 m, the initial potential energy is PE_initial = m × g × h_initial, where h_initial = 2.0 m.
2. Determine the potential energy after the bounce: The ball bounces back to a height of 1.5 m. So the potential energy after the bounce is PE_final = m × g × h_final, where h_final = 1.5 m.
3. Calculate the fraction of energy lost: The fraction of energy lost is given by (PE_initial - PE_final) / PE_initial.
So, the fraction of the ball's initial energy lost during the bounce is [(PE_initial - PE_final) / PE_initial].
(b) Speed of the ball just as it leaves the ground after the bounce:
To find the speed of the ball just as it leaves the ground after the bounce, we can use the conservation of mechanical energy. At the highest point of the bounce, the ball's total mechanical energy is equal to the initial potential energy. This energy is later converted to kinetic energy just as it leaves the ground. We can calculate it using the following steps:
1. Calculate the potential energy at the maximum height: PE_max = m × g × h_final, where h_final = 1.5 m.
2. Equate the potential energy at the maximum height to the kinetic energy just as it leaves the ground. The equation is: KE = 0.5 × m × v^2, where KE is the kinetic energy, m is the mass of the ball, and v is the velocity/speed.
So, 0.5 × m × v^2 = PE_max.
3. Rearrange the equation to solve for v: v = √(2 × (PE_max / m)).
Therefore, the ball's speed just as it leaves the ground after the bounce is given by v = √(2 × (PE_max / m)).
To solve this problem, we can use the principle of energy conservation. The initial energy of the ball is equal to its potential energy at the starting height, and the final energy is equal to the potential energy at the highest point during the bounce.
(a) To find the fraction of the initial energy lost during the bounce, we need to compare the difference in potential energy before and after the bounce.
Initial potential energy = mgh1
Final potential energy = mgh2
Given:
Height before bounce, h1 = 2.0 m
Height after bounce, h2 = 1.5 m
The fraction of energy lost can be calculated using the following formula:
Fraction of energy lost = (Initial potential energy - Final potential energy) / Initial potential energy
Let's plug in the values and calculate:
Fraction of energy lost = (mgh1 - mgh2) / mgh1
Fraction of energy lost = (2.0 - 1.5) / 2.0
Fraction of energy lost = 0.5 / 2.0
Fraction of energy lost = 0.25
Therefore, the fraction of the initial energy lost during the bounce is 0.25 or 25%.
(b) To find the ball's speed just as it leaves the ground after the bounce, we can use the conservation of mechanical energy:
Initial potential energy = Final kinetic energy
The initial potential energy is equal to mgh1, and the final kinetic energy is equal to (1/2)mv^2, where v is the speed of the ball.
mgh1 = (1/2)mv^2
Cross-multiplying and canceling the mass:
2gh1 = v^2
Now, let's plug in the values and solve for v:
v^2 = 2 * 9.8 * 2.0
v^2 = 39.2
v ≈ √39.2
v ≈ 6.26 m/s
Therefore, the ball's speed just as it leaves the ground after the bounce is approximately 6.26 m/s.