Trying to determine its depth, a rock climber drops a pebble into a chasm and hears the pebble strike the ground 4.00 s later.
(a) If the speed of sound in air is 343 m/s at the rock climber's location, what is the depth of the chasm?

(b) What is the percentage of error that would result from assuming the speed of sound is infinite?

Let's solve this step by step.

### Part (a) - Determining the depth of the chasm

Let:
- \( d \) = depth of the chasm (in meters),
- \( g \) = acceleration due to gravity = 9.8 m/s²,
- \( t_{total} \) = total time until the climber hears the sound = 4.00 s,
- \( v_s \) = speed of sound = 343 m/s.

The total time \( t_{total} \) is the sum of two times:
1. \( t_1 \) = the time it takes for the pebble to fall to the bottom.
2. \( t_2 \) = the time it takes for the sound to travel back up.

So, the equation for the total time is:

\[
t_{total} = t_1 + t_2
\]

Now let's analyze each time:
1. For the pebble's fall, we use the equation of motion:
\[
d = \frac{1}{2} g t_1^2 \quad \Rightarrow \quad t_1 = \sqrt{\frac{2d}{g}}
\]
2. For the sound to travel back up, it takes:
\[
t_2 = \frac{d}{v_s}
\]

Substituting these into the total time equation:
\[
t_{total} = \sqrt{\frac{2d}{g}} + \frac{d}{v_s}
\]

Now, solve for \( d \) using the given total time of 4.00 s. I'll calculate that.

---

### Part (b) - Error with Infinite Speed of Sound

If the speed of sound were assumed to be infinite, the sound would be heard instantaneously, so \( t_2 = 0 \), and the total time would only depend on the time for the pebble to fall:

\[
t_{total} = t_1 = \sqrt{\frac{2d}{g}}
\]

Solve for \( d \) under this assumption. Then, compare the result with the actual depth found in part (a) to determine the percentage error:

\[
\text{Percentage Error} = \frac{|d_{\infty} - d_{\text{actual}}|}{d_{\text{actual}}} \times 100\%
\]

I'll calculate both parts.### Part (a) - Depth of the Chasm:
The depth of the chasm is approximately \( 70.54 \, \text{m} \).

### Part (b) - Percentage Error with Infinite Speed of Sound:
If we assume the speed of sound is infinite, the depth would be calculated as \( 78.4 \, \text{m} \). The percentage error resulting from this assumption is approximately \( 11.13\% \).

Would you like further details or clarifications on any of the steps?

Here are 5 related questions for further exploration:
1. How would the result change if the speed of sound were different, such as at higher altitudes?
2. What is the effect of air resistance on the pebble's fall, and how could it be accounted for?
3. How would the depth calculation change if the pebble were thrown downward with an initial velocity?
4. Can this method be applied to measuring the depth of a well filled with water instead of air?
5. How does the speed of sound vary with temperature, and how would that affect this calculation?

**Tip:** In many real-world problems, accounting for delays like the speed of sound or air resistance can significantly impact the accuracy of physical measurements.

Trying to determine its depth, a rock climber drops a pebble into a chasm and hears the pebble strike the ground 4.00 s later. (a) If the speed of sound in air is 343 m/s at the rock climber's location, what is the depth of the chasm? (b) What is the percentage of error that would result from assuming the speed of sound is infinite?

Explore how to calculate the depth of a chasm using kinematic equations and sound propagation. The problem involves dropping a pebble into a chasm and analyzing the time it takes for the sound to travel back to determine the depth accurately. Learn how assuming an infinite speed of sound can introduce a percentage error in the calculation. Suitable for Grades 11-12 students.

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