https://file.aiquickdraw.com/mathbot/file/1728811395116ye5ilrut.jpg

Let's break down and solve each problem.

### 1. **Potential energy of a lifted box**

The formula for gravitational potential energy is:
\[
PE = mgh
\]
Where:
- \( PE = 176 \, \text{J} \) (given),
- \( m = 6.2 \, \text{kg} \) (mass of the box),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( h \) is the height of the shelf.

We rearrange the equation to solve for \( h \):
\[
h = \frac{PE}{mg} = \frac{176}{6.2 \times 9.8}
\]
\[
h = \frac{176}{60.76} \approx 2.9 \, \text{m}
\]
So, the shelf is approximately **2.9 meters** high.

---

### 2. **Mass of the man climbing a wall**

Using the same potential energy formula:
\[
PE = mgh
\]
Where:
- \( PE = 3168 \, \text{J} \),
- \( h = 5.2 \, \text{m} \),
- \( g = 9.8 \, \text{m/s}^2 \).

Rearranging to solve for \( m \):
\[
m = \frac{PE}{gh} = \frac{3168}{9.8 \times 5.2}
\]
\[
m = \frac{3168}{50.96} \approx 62.2 \, \text{kg}
\]
So, the mass of the man is approximately **62.2 kg**.

---

### 3. **Potential energy of the ball on a slope**

First, we need to calculate the vertical height of the ball using trigonometry. The right triangle has:
- Opposite side (height) = 20 cm = 0.2 m,
- Hypotenuse = 50 cm = 0.5 m.

The angle \( \theta \) can be calculated by:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{0.2}{0.5} = 0.4
\]
Now, using potential energy:
\[
PE = mgh
\]
Where:
- \( m = 907 \, \text{g} = 0.907 \, \text{kg} \),
- \( g = 9.8 \, \text{m/s}^2 \),
- \( h = 0.2 \, \text{m} \).

\[
PE = 0.907 \times 9.8 \times 0.2 \approx 1.78 \, \text{J}
\]
The ball gains approximately **1.78 J** of potential energy.

---

### 4. **Kinetic energy of a moving car**

The formula for kinetic energy is:
\[
KE = \frac{1}{2} mv^2
\]
Where:
- \( m = 805 \, \text{kg} \),
- \( v = 26 \, \text{m/s} \).

\[
KE = \frac{1}{2} \times 805 \times 26^2 = \frac{1}{2} \times 805 \times 676 = 272,690 \, \text{J}
\]
So, the kinetic energy of the car is approximately **272,690 J** (or **272.69 kJ**).

---

### 5. **Kinetic energy of a cyclist and bike**

Using the same kinetic energy formula:
\[
KE = \frac{1}{2} mv^2
\]
Where:
- \( m = 111 \, \text{kg} \) (total mass of cyclist and bike),
- \( v = 12 \, \text{m/s} \).

\[
KE = \frac{1}{2} \times 111 \times 12^2 = \frac{1}{2} \times 111 \times 144 = 7,992 \, \text{J}
\]
The kinetic energy is approximately **7,992 J** (or **7.99 kJ**).

---

### 6. **Mass of a tennis ball based on kinetic energy**

Rearrange the kinetic energy formula to solve for mass:
\[
KE = \frac{1}{2} mv^2 \quad \text{so} \quad m = \frac{2KE}{v^2}
\]
Where:
- \( KE = 72 \, \text{J} \),
- \( v = 53 \, \text{m/s} \).

\[
m = \frac{2 \times 72}{53^2} = \frac{144}{2809} \approx 0.0512 \, \text{kg}
\]
The mass of the tennis ball is approximately **0.0512 kg** (or **51.2 g**).

---

Would you like further details on any of these solutions? 

Here are 5 follow-up questions related to these problems:

1. How would the potential energy change if the box's mass doubled in problem 1?
2. What effect does increasing the height of the wall in problem 2 have on potential energy?
3. How does the angle of the slope affect the potential energy gain in problem 3?
4. If the car's speed was doubled, how would its kinetic energy change?
5. How does the kinetic energy of the cyclist and bike change if they slow down to 6 m/s?

**Tip:** Always keep track of units in physics problems to avoid calculation errors.

Solve the following problems:
1. A box has a mass of 6.2 kg. The box is lifted from the garage floor and placed on a shelf. If the box gains 176 J of Potential Energy, how high is the shelf?
2. A man climbs onto a wall that is 5.2 m high and gains 3168J of potential energy. What is the mass of the man?
3. A 907 g ball is pulled up a slope as shown in the diagram. Calculate the potential energy it gains.
4. A car with a mass of 805 kg is moving with a speed of 26 m/s. Calculate the kinetic energy of the car.
5. A cyclist and bike have a total mass of 111 kg and a speed of 12 m/s. Calculate the kinetic energy.
6. A tennis ball is traveling at 53 m/s and has a kinetic energy of 72 J. Calculate the mass of the tennis ball.

Solve a set of physics problems involving potential and kinetic energy. Learn to calculate the height of a shelf based on potential energy, the mass of a man based on energy gain, the energy gained by a ball on a slope, and more. Suitable for students in Grades 9-12.

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