A 10 kg bowling ball sits at the top of a 10 m hill and then slides down its icy hillside.
a. What is the speed of the bowling ball when it reaches the bottom of the hill?
b. What is the change in kinetic energy of the system as the bowling ball travels
from the top of the hill to the bottom of the hill?
c. What is the bowling ball’s mechanical energy at the top of the hill?
d. What is the bowling ball’s mechanical energy at the bottom of the hill?
a. conservation of energy, KEbottom=PE top
1/2 m v^2=mgh solve for v.
b,c,d In the physical sciences, mechanical energy is the sum of potential energy and kinetic energy. It is the energy associated with the motion and position of an object.
To answer these questions, we need to make a few assumptions:
1. We will neglect any friction or air resistance.
2. We will consider the bowling ball as a point mass.
3. We will assume the hill is a straight slope (no curves or bumps).
Now, let's address each question step-by-step:
a. To find the speed of the bowling ball at the bottom of the hill, we can use the principle of conservation of energy. The mechanical energy at the top of the hill (PE + KE) is equal to the mechanical energy at the bottom of the hill.
The mechanical energy at the top of the hill is given by the potential energy, which can be calculated using the formula:
PE = mgh
where m is the mass of the ball (10 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hill (10 m). Substituting the values, we get:
PE = (10 kg)(9.8 m/s^2)(10 m) = 980 J
At the bottom of the hill, all of the potential energy is converted into kinetic energy. Therefore, the kinetic energy can be calculated using the formula:
KE = 1/2 mv^2
where v is the speed of the ball. Substituting the values, we get:
980 J = 1/2 (10 kg) v^2
v^2 = (2 * 980 J) / 10 kg
v^2 = 196
v = √196
v ≈ 14 m/s
So, the speed of the bowling ball when it reaches the bottom of the hill is approximately 14 m/s.
b. The change in kinetic energy of the system as the bowling ball travels from the top of the hill to the bottom is equal to the difference between the kinetic energy at the bottom and the kinetic energy at the top.
The initial kinetic energy at the top of the hill is zero since the ball is at rest. The final kinetic energy at the bottom of the hill can be calculated using the formula:
KE = 1/2 mv^2
where m is the mass of the ball (10 kg) and v is the velocity of the ball (14 m/s). Substituting the values, we get:
KE = 1/2 (10 kg) (14 m/s)^2
KE = 1/2 (10 kg) (196 m^2/s^2) = 980 J
The change in kinetic energy is therefore:
ΔKE = KE_final - KE_initial
ΔKE = 980 J - 0 J = 980 J
Thus, the change in kinetic energy of the system as the ball travels from the top of the hill to the bottom is 980 J.
c. The bowling ball's mechanical energy at the top of the hill includes the potential energy and the kinetic energy. As calculated earlier, the potential energy is 980 J. Since the bowling ball is at rest at the top of the hill, the initial kinetic energy is zero. Therefore, the mechanical energy at the top of the hill is:
Mechanical energy at the top = Potential energy + Kinetic energy
Mechanical energy at the top = 980 J + 0 J = 980 J
So, the bowling ball's mechanical energy at the top of the hill is 980 J.
d. The bowling ball's mechanical energy at the bottom of the hill also includes the potential energy and the kinetic energy. As calculated earlier, all the potential energy is converted into kinetic energy at the bottom of the hill, which is 980 J. Therefore, the mechanical energy at the bottom of the hill is:
Mechanical energy at the bottom = Potential energy + Kinetic energy
Mechanical energy at the bottom = 0 J + 980 J = 980 J
So, the bowling ball's mechanical energy at the bottom of the hill is 980 J.
a. To find the speed of the bowling ball when it reaches the bottom of the hill, we can use the principle of conservation of energy. At the top of the hill, the bowling ball only has potential energy, and at the bottom of the hill, it will have both potential and kinetic energy.
To calculate the speed, we can equate the initial potential energy at the top of the hill to the final kinetic energy at the bottom of the hill. The potential energy (PE) of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
At the top of the hill:
PE = mgh = 10 kg * 9.8 m/s² * 10 m = 980 J (joules)
At the bottom of the hill, all the potential energy is converted into kinetic energy. The kinetic energy (KE) of an object is given by the formula KE = 1/2 * mv², where v is the final velocity.
So, setting the potential energy equal to the kinetic energy:
PE = KE
980 J = 1/2 * 10 kg * v²
v² = (2 * 980 J) / 10 kg
v² = 196 m²/s²
Taking the square root of both sides, we find the speed:
v = √(196 m²/s²) ≈ 14 m/s
Therefore, the speed of the bowling ball when it reaches the bottom of the hill is approximately 14 m/s.
b. The change in kinetic energy (ΔKE) of the system as the bowling ball travels from the top of the hill to the bottom of the hill can be found by subtracting the initial kinetic energy of the system from the final kinetic energy.
At the top of the hill, the bowling ball is at rest, so the initial kinetic energy (KEi) is 0 J.
At the bottom of the hill, we found in part a that the bowling ball has a speed of 14 m/s. Therefore, the final kinetic energy (KEf) is given by the formula KE = 1/2 * mv²:
KEf = 1/2 * 10 kg * (14 m/s)²
KEf = 1/2 * 10 kg * 196 m²/s²
KEf = 980 J
Now we can calculate the change in kinetic energy:
ΔKE = KEf - KEi
ΔKE = 980 J - 0 J
ΔKE = 980 J
Therefore, the change in kinetic energy of the system as the bowling ball travels from the top of the hill to the bottom of the hill is 980 J.
c. The bowling ball's mechanical energy at the top of the hill consists of its potential energy (PE) and kinetic energy (KE) if any. Since the ball is at rest at the top of the hill, its kinetic energy is 0 J. Therefore, the mechanical energy at the top of the hill is equal to the potential energy:
ME = PE = 980 J
So, the bowling ball's mechanical energy at the top of the hill is 980 J.
d. At the bottom of the hill, the bowling ball has both potential energy and kinetic energy. We found in part a that the kinetic energy at the bottom of the hill is 980 J.
The mechanical energy at the bottom of the hill is the sum of the potential energy and kinetic energy:
ME = PE + KE = 0 J + 980 J
Therefore, the bowling ball's mechanical energy at the bottom of the hill is 980 J.