Question

a restraint is keeping two weights at the same height in a pulley system. the mass of m1 is greater than the mass of m2. once the restraint is released, the weights are able to move, and energy is transformed in the system. check the box to correctly match the description with the type of energy present in the system or the type of transformation that is occurring. potential energy kinetic energy to potential energy potential energy to kinetic energy energy of m1 before the restraint is released energy transformation of m1 as it begins to fall energy transformation of m2 as it rises a 1 kg ball is thrown straight up into the air and its height is recorded every second until it reaches the ground a little more than 3 seconds later. the height of the ball above the ground is shown in the table. what is the potential energy of the ball at each time? gravitational potential energy can be calculated by multiplying the mass, the height, and the acceleration due to gravity (9.8 m/s²). at what time would the ball have the highest gravitational potential energy for the ball be

Explanation:

Part 1: Pulley System Energy Matching

1. Energy of M1 before restraint release

• Answer: potential energy
• Brief Explanations: Before release, M1 is at rest (no motion, so no kinetic energy) but has height (gravitational potential energy depends on height and mass). Since it’s stationary, its energy is potential.

2. Energy transformation of M1 as it falls

• Answer: potential energy to kinetic energy
• Brief Explanations: As M1 falls, its height decreases (losing potential energy) and its speed increases (gaining kinetic energy). So potential energy converts to kinetic energy.

3. Energy transformation of M2 as it rises

• Answer: kinetic energy to potential energy
• Brief Explanations: M2 rises because M1 pulls it. M2 gains height (potential energy) as it moves (kinetic energy from the system’s motion). So kinetic energy (from movement) converts to potential energy (from height).

Part 2: Gravitational Potential Energy of the Ball

(Note: The table with height vs. time is missing, but we use the formula \( PE = mgh \), where \( m = 1 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( h \) is height at each time.)

Step 1: Recall the formula for gravitational potential energy

The formula is \( PE = m \times g \times h \), where:

• \( m = 1 \, \text{kg} \) (mass of the ball),
• \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
• \( h \) = height (from the table, e.g., if at time \( t \), height is \( h_t \), then \( PE_t = 1 \times 9.8 \times h_t = 9.8h_t \) joules).

Step 2: Highest Potential Energy

Gravitational potential energy \( PE \) is proportional to height (\( PE \propto h \)) because \( m \) and \( g \) are constant. Thus, the ball has the highest \( PE \) at the time when its height is maximum (since \( PE = 9.8h \), larger \( h \) means larger \( PE \)).

Example (If Table Were Provided):

Suppose the table has heights:

• Time 0: \( h = 0 \, \text{m} \) → \( PE = 1 \times 9.8 \times 0 = 0 \, \text{J} \)
• Time 1: \( h = 5 \, \text{m} \) → \( PE = 1 \times 9.8 \times 5 = 49 \, \text{J} \)
• Time 2: \( h = 8 \, \text{m} \) → \( PE = 1 \times 9.8 \times 8 = 78.4 \, \text{J} \)
• Time 3: \( h = 3 \, \text{m} \) → \( PE = 1 \times 9.8 \times 3 = 29.4 \, \text{J} \)

Then, the highest \( PE \) is at Time 2 (when height is 8 m).

Final Answers (Pulley System):

1. Energy of M1 before release: \(\boldsymbol{\text{potential energy}}\)
2. M1’s energy transformation: \(\boldsymbol{\text{potential energy to kinetic energy}}\)
3. M2’s energy transformation: \(\boldsymbol{\text{kinetic energy to potential energy}}\)

(For the ball’s \( PE \), apply \( PE = 9.8h \) using the table’s height values, and the highest \( PE \) occurs at the time with the maximum height.)

Answer:

Part 1: Pulley System Energy Matching

1. Energy of M1 before restraint release

• Answer: potential energy
• Brief Explanations: Before release, M1 is at rest (no motion, so no kinetic energy) but has height (gravitational potential energy depends on height and mass). Since it’s stationary, its energy is potential.

2. Energy transformation of M1 as it falls

• Answer: potential energy to kinetic energy
• Brief Explanations: As M1 falls, its height decreases (losing potential energy) and its speed increases (gaining kinetic energy). So potential energy converts to kinetic energy.

3. Energy transformation of M2 as it rises

• Answer: kinetic energy to potential energy
• Brief Explanations: M2 rises because M1 pulls it. M2 gains height (potential energy) as it moves (kinetic energy from the system’s motion). So kinetic energy (from movement) converts to potential energy (from height).

Part 2: Gravitational Potential Energy of the Ball

(Note: The table with height vs. time is missing, but we use the formula \( PE = mgh \), where \( m = 1 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( h \) is height at each time.)

Step 1: Recall the formula for gravitational potential energy

The formula is \( PE = m \times g \times h \), where:

• \( m = 1 \, \text{kg} \) (mass of the ball),
• \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
• \( h \) = height (from the table, e.g., if at time \( t \), height is \( h_t \), then \( PE_t = 1 \times 9.8 \times h_t = 9.8h_t \) joules).

Step 2: Highest Potential Energy

Gravitational potential energy \( PE \) is proportional to height (\( PE \propto h \)) because \( m \) and \( g \) are constant. Thus, the ball has the highest \( PE \) at the time when its height is maximum (since \( PE = 9.8h \), larger \( h \) means larger \( PE \)).

Example (If Table Were Provided):

Suppose the table has heights:

• Time 0: \( h = 0 \, \text{m} \) → \( PE = 1 \times 9.8 \times 0 = 0 \, \text{J} \)
• Time 1: \( h = 5 \, \text{m} \) → \( PE = 1 \times 9.8 \times 5 = 49 \, \text{J} \)
• Time 2: \( h = 8 \, \text{m} \) → \( PE = 1 \times 9.8 \times 8 = 78.4 \, \text{J} \)
• Time 3: \( h = 3 \, \text{m} \) → \( PE = 1 \times 9.8 \times 3 = 29.4 \, \text{J} \)

Then, the highest \( PE \) is at Time 2 (when height is 8 m).

Final Answers (Pulley System):

1. Energy of M1 before release: \(\boldsymbol{\text{potential energy}}\)
2. M1’s energy transformation: \(\boldsymbol{\text{potential energy to kinetic energy}}\)
3. M2’s energy transformation: \(\boldsymbol{\text{kinetic energy to potential energy}}\)

(For the ball’s \( PE \), apply \( PE = 9.8h \) using the table’s height values, and the highest \( PE \) occurs at the time with the maximum height.)