A spelunker (cave explorer) drops a stone from rest into a hole. The speed of sound is 343 m/s in air, and the sound of the stone striking the bottom is heard 3.24 s after the stone is dropped. How deep is the hole?

Well, let's calculate that, shall we? Since the speed of sound is 343 m/s and it takes 3.24 seconds for the sound to reach the spelunker, we can use the formula distance = speed × time. Therefore, the depth of the hole would be 343 m/s × 3.24 s = 1111.32 meters. So, it seems the spelunker has quite a deep hole to deal with!

To determine the depth of the hole, we need to analyze the time it takes for the sound to travel back to the surface.

1. First, we calculate the time it takes for the stone to hit the bottom of the hole, considering it drops from rest:
- We know that the acceleration due to gravity is approximately 9.8 m/s^2.
- The time it takes for the stone to hit the bottom can be found using the equation s = 0.5 * g * t^2, where s is the distance (or depth of the hole) and t is the time.
- Rearranging the equation, we get t = square root of (2s / g).

2. Next, we calculate the time sound takes to reach the surface:
- The sound travels down to the bottom of the hole and then reflects back to the top.
- Therefore, it covers the distance twice.
- The time it takes for the sound to travel back to the surface is given as 3.24 s.

3. Equating the time for the stone to hit the bottom and the time for the sound to travel back to the surface, we have:
square root of (2s / g) = 3.24 s

4. Solving for s (the depth of the hole):
- To isolate s, square both sides of the equation:
2s / g = (3.24 s)^2

- Multiply both sides by g:
2s = g * (3.24 s)^2

- Divide both sides by 2:
s = 0.5 * g * (3.24 s)^2

5. Finally, calculate s using the values provided:
- Plug in the value for g (approximately 9.8 m/s^2):
s = 0.5 * 9.8 * (3.24 s)^2

- Evaluate the equation:
s ≈ 0.5 * 9.8 * (10.4976 s^2)

s ≈ 51.086 m

Therefore, the depth of the hole is approximately 51.086 meters.

To find the depth of the hole, we need to calculate the time it takes for the stone to travel to the bottom and back up, and then use the speed of sound to determine the total distance traveled.

First, let's calculate the time it takes for the stone to hit the bottom. The sound is heard 3.24 seconds after the stone is dropped. Since the sound travels at the speed of sound in air, which is 343 m/s, the time it takes for the sound to travel from the bottom to the explorer's ears is the same as the time it takes for the stone to fall. Therefore, the time it takes for the stone to hit the bottom is also 3.24 seconds.

Next, we need to find the distance the stone falls. We can use the formula for distance traveled under constant acceleration:

d = 0.5 * g * t^2

where d is the distance, g is the acceleration due to gravity (which is approximately 9.8 m/s^2), and t is the time.

Plugging in the values, we get:

d = 0.5 * 9.8 m/s^2 * (3.24 s)^2
≈ 0.5 * 9.8 m/s^2 * 10.4976 s^2
≈ 51.492 m

So, the depth of the hole is approximately 51.492 meters.

t1 + t2 = 3.24s.

t2 = 3.24 - t1.

t1 = Flight time of stone.
t2 = Travel time of the sound.

d1 = 0.5gt^2 = distance the stone traveled.

d2 = Vt = Distance the sound traveled.

d1 = d2,
0.5*9.8t^2 = 343m/s * t2,
Substitute 3.24 - t1 for t2:
4.9t1^2 = 343 * (3.24 - t1),
4.9t1^2 = 1111.3 - 343t1,
4.91t1^2 + 343t1 - 1111.3 = 0,
Use Quadratic Formula to find t and get:
t1 = 3.10244, and -73.10.
Use the positive value of t.
t1 = 3.10244s = time in flight for stone.

d1 = 0.5 * 9.8 * (3.10244)^2,
d1 = 47.2m = Depth of hole.