How High is the Cliff if a Rock Splashes Water in 3 Seconds?

[Jerry]

Homework Statement

"A rock is dropped off a cliff into the water below. The sound of the splash is heard 3.0 s later. If the speed of sound is 332 m/s, calculate the height of the cliff above the water. (Note: the total time it takes for the rock to fall and the sound to travel upwards is 3.0 s)"

Therefore,
v₁ = 0
g = 9.8 m/s²
Δt = 3.0 s
v_sound = 332 m/s

Homework Equations

FOR SOUND
Δd = v_sound * Δt₂, where Δt₂ is the time it takes from the sound to reach the top of the cliff from the bottom.
FOR ROCK
Δd = v₁ * Δt + 0.5 * g * (Δt)²
*the following equations may be useful but i doubt it*
v₂ = v₁ + g * Δt
(v₂)² = (v₁)² + 2 * g * Δd

*assume no air resistance

The Attempt at a Solution

Δd = ?
No clue.

I figured that the answer should be 41 m, I believe, by trial and error but I would like to know how this can be solved in a normal way.

 

[Simon Bridge]

The Δt for the rock is different from the Δt you set for the total time.
Call it Δt₁ ?

Then Δt₁+Δt₂=what?

Now you have three equations and three unknowns.

 

[Jerry]

It should be time for rock + time for sound = 3.0s and that the sound is not affected by gravity but rock is accelerating from 0 , at 9.8m/s².

 

[Simon Bridge]

Well done - now you can solve it.
hint: all three equations have to be true simultaneously.

 

[HalfLostAndSearching]

I would like to clarify a few things before providing a response to the given problem. The problem states that the sound of the splash is heard 3.0 seconds later, which means that the total time for the rock to fall and the sound to travel upwards is 3.0 seconds. This is important information because it indicates that the sound travels both downwards and upwards in this scenario. Additionally, the given equations for sound and rock are not applicable to this problem as they do not take into account the sound traveling both ways.

To solve this problem, we can use the following equation for the distance traveled by sound:

Δd = 0.5 * vsound * Δt

Since the total time for the sound to travel both ways is 3.0 seconds, we can divide this time by 2 to get the time it takes for the sound to travel from the bottom of the cliff to the top, which is 1.5 seconds. We can then plug in the given speed of sound (332 m/s) and the time (1.5 s) into the equation to get the distance traveled by sound:

Δd = 0.5 * 332 m/s * 1.5 s = 249 m

This distance represents the total vertical distance that the sound travels, which is equal to the height of the cliff. Therefore, the height of the cliff above the water is 249 meters.

In conclusion, this problem can be solved by using the equation for the distance traveled by sound and taking into account the total time it takes for the sound to travel both ways. It is important to carefully consider all the given information and use the appropriate equations to solve the problem accurately.