Suppose 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 and, 14.0s later, hear the sound of it hitting the ground at the foot of the cliff. Ignoring air resistance, how high is the cliff if the speed of sound is 330m/s?

To solve this problem, we need to break it into two parts:

1. **The time it takes for the rock to fall to the ground.**
2. **The time it takes for the sound to travel back to the top of the cliff.**

### Step 1: Break down the total time
Let:
- \( t_1 \) be the time it takes for the rock to fall.
- \( t_2 \) be the time it takes for the sound to travel back up.

We know:
- Total time, \( t_1 + t_2 = 14.0 \) seconds.
- The speed of sound is \( 330 \, \text{m/s} \).

### Step 2: Set up equations

#### For the rock falling:
We can use the equation of motion for free fall:
\[
h = \frac{1}{2} g t_1^2
\]
Where:
- \( h \) is the height of the cliff.
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity).
- \( t_1 \) is the time for the rock to fall.

#### For the sound:
The time for the sound to travel back up is:
\[
t_2 = \frac{h}{330}
\]

### Step 3: Solve the system of equations

The total time is the sum of both times:
\[
t_1 + t_2 = 14.0 \, \text{s}
\]
Substitute \( t_2 = \frac{h}{330} \):
\[
t_1 + \frac{h}{330} = 14.0
\]

Now, express \( h \) in terms of \( t_1 \):
\[
h = \frac{1}{2} g t_1^2 = 4.9 t_1^2
\]
Substitute this into the previous equation:
\[
t_1 + \frac{4.9 t_1^2}{330} = 14.0
\]
Now, let's solve this equation to find \( t_1 \).The valid solution for \( t_1 \) (time it takes the rock to fall) is approximately \( 11.90 \, \text{seconds} \) (we discard the negative value since time cannot be negative).

### Step 4: Calculate the height of the cliff

Now that we have \( t_1 = 11.90 \, \text{seconds} \), we can use the equation for the height of the cliff:
\[
h = 4.9 \times t_1^2
\]
Let's compute \( h \).The height of the cliff is approximately \( 693.66 \, \text{meters} \).

Would you like further details or clarification?

Here are 5 related questions for further exploration:
1. How would air resistance affect the time for the rock to fall?
2. What would happen if the speed of sound was different, say on another planet?
3. How does gravity influence the time it takes for an object to fall from a height?
4. How could you estimate the speed of sound using this kind of experiment?
5. How would the calculation change if the total time was measured inaccurately?

**Tip**: When solving problems involving motion and sound, always account for delays introduced by sound traveling at finite speeds!

Learn how to calculate the height of a cliff by analyzing the time it takes for a rock to fall and the sound to travel back. This problem involves concepts of free fall, the speed of sound, and kinematic equations. Suitable for high school students in Grades 9-12.

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