Question
A ball is dropped from rest from point 1, and by the time it reaches point 2, it has gained 2 J of kinetic energy. As it continues its fall from point 2 to point 3, its kinetic energy increases by another 6 J.
Image
What is the ball's change in gravitational potential energy as it falls from point 1 to point 3?
a
a decrease of about 6 J
b
an increase of about 6
c
a decrease of about 8 J
d
an increase of about 8 J
Image
What is the ball's change in gravitational potential energy as it falls from point 1 to point 3?
a
a decrease of about 6 J
b
an increase of about 6
c
a decrease of about 8 J
d
an increase of about 8 J
Answers
GPT-4o mini
To determine the ball's change in gravitational potential energy as it falls from point 1 to point 3, we can use the principle of conservation of energy. The total mechanical energy (kinetic energy + potential energy) of the ball remains constant when only gravitational forces are considered.
1. Initially, at point 1, the ball has some gravitational potential energy (PE_initial) and no kinetic energy (KE_initial = 0).
2. When it reaches point 2, the ball has gained 2 J of kinetic energy (KE_final at point 2 = 2 J). The change in kinetic energy from point 1 to point 2 is +2 J.
3. As it continues to fall from point 2 to point 3, the ball gains an additional 6 J of kinetic energy, so its total kinetic energy at point 3 is KE_final at point 3 = 2 J + 6 J = 8 J.
Now, we can calculate the total change in kinetic energy from point 1 to point 3:
- The change in kinetic energy from point 1 to point 3 is \(8 J - 0 J = 8 J\).
According to the conservation of energy:
\[
\Delta PE + \Delta KE = 0
\]
Thus, the change in gravitational potential energy (ΔPE) must equal the negative of the change in kinetic energy (ΔKE):
\[
\Delta PE = -\Delta KE
\]
Given that ΔKE is +8 J (the kinetic energy increased by 8 J from point 1 to point 3):
\[
\Delta PE = -8 J
\]
This indicates a decrease in gravitational potential energy of 8 J.
Therefore, the answer is:
**c. a decrease of about 8 J**.
1. Initially, at point 1, the ball has some gravitational potential energy (PE_initial) and no kinetic energy (KE_initial = 0).
2. When it reaches point 2, the ball has gained 2 J of kinetic energy (KE_final at point 2 = 2 J). The change in kinetic energy from point 1 to point 2 is +2 J.
3. As it continues to fall from point 2 to point 3, the ball gains an additional 6 J of kinetic energy, so its total kinetic energy at point 3 is KE_final at point 3 = 2 J + 6 J = 8 J.
Now, we can calculate the total change in kinetic energy from point 1 to point 3:
- The change in kinetic energy from point 1 to point 3 is \(8 J - 0 J = 8 J\).
According to the conservation of energy:
\[
\Delta PE + \Delta KE = 0
\]
Thus, the change in gravitational potential energy (ΔPE) must equal the negative of the change in kinetic energy (ΔKE):
\[
\Delta PE = -\Delta KE
\]
Given that ΔKE is +8 J (the kinetic energy increased by 8 J from point 1 to point 3):
\[
\Delta PE = -8 J
\]
This indicates a decrease in gravitational potential energy of 8 J.
Therefore, the answer is:
**c. a decrease of about 8 J**.