You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.20 s
later you hear the sound of the rock hitting the ground at the foot of the cliff. If you ignore air resistance, how high is the cliff if the speed of sound is 330  m/s
?

To find the height of the cliff, we'll break the problem into two parts: 

1. **Time taken by the rock to fall** from the top of the cliff to the bottom.
2. **Time taken by the sound to travel back up** from the bottom to the top.

Given:
- Total time for the rock to fall and the sound to travel back: \( t_{\text{total}} = 8.20 \) seconds
- Speed of sound: \( v_{\text{sound}} = 330 \) m/s
- Acceleration due to gravity: \( g = 9.81 \) m/s²

### Step 1: Define the unknowns

Let:
- \( h \) be the height of the cliff.
- \( t_1 \) be the time it takes for the rock to fall.
- \( t_2 \) be the time it takes for the sound to travel back up.

From the total time:
\[
t_{\text{total}} = t_1 + t_2
\]
\[
t_2 = \frac{h}{v_{\text{sound}}}
\]
\[
t_1 + \frac{h}{330} = 8.20
\]

### Step 2: Use the equation of motion for the rock's fall

The distance fallen by the rock is given by:
\[
h = \frac{1}{2} g t_1^2
\]
\[
h = \frac{1}{2} \times 9.81 \times t_1^2
\]

### Step 3: Substitute \( t_2 \) into the total time equation

Substituting \( h \) from the rock’s fall equation into the sound travel time equation:
\[
t_1 + \frac{\frac{1}{2} \times 9.81 \times t_1^2}{330} = 8.20
\]

### Step 4: Solve for \( t_1 \) and then find \( h \)

We have the equation:
\[
t_1 + \frac{4.905 \times t_1^2}{330} = 8.20
\]
This equation can be solved using algebraic or numerical methods.

Let's calculate \( t_1 \) and then use it to find \( h \).

We can solve this equation using Python.The two solutions for \( t_1 \) (the time it takes for the rock to fall) are approximately:

- \( t_1 \approx -74.67 \) seconds (not physically meaningful since time cannot be negative)
- \( t_1 \approx 7.39 \) seconds

So, \( t_1 = 7.39 \) seconds is the valid solution.

### Step 5: Calculate the height of the cliff \( h \)

Now that we have \( t_1 \), we can calculate the height \( h \) using the formula:
\[
h = \frac{1}{2} g t_1^2
\]
Let's calculate \( h \).The height of the cliff is approximately \( 267.77 \) meters.

Would you like more details, or do you have any questions? Here are a few related questions you might find interesting:

1. How would the height change if the speed of sound was different?
2. How would air resistance affect the time taken for the rock to fall?
3. What would be the height if the total time was 10 seconds instead of 8.20 seconds?
4. Can you derive a general formula for the height in terms of total time and speed of sound?
5. How would the height calculation differ on another planet with different gravity?

**Tip:** Always double-check the units in physics problems to ensure accuracy.

Learn how to calculate the height of a cliff using the time taken for a rock to fall and the speed of sound. This physics problem involves kinematics and the principles of projectile motion. Find out how air resistance and different speeds of sound affect the calculation. Suitable for high school students.

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