A rock is dropped into a well, and a splash was heard 3 seconds later. If sound travels in air at 344 m/s,
A) How deep is the well?
B) What are the final positions of the Rock and the Sound? (Rock = Yfr; Sound = Yfs)
Known Variables:
Gravity = -9 m/s^2 (Rock falling is Free Fall)
Rock = Yr, Rock Initial Position = Yir
Sound = Ys, Sound Initial Position = Yis
Edit: Gravity = -9.8 m/s^2
To solve this problem, we need to use the equations of motion and the principles of free fall to find the depth of the well and the final positions of the rock and the sound.
A) How deep is the well?
To find the depth of the well, we need to determine the time it takes for the rock to reach the bottom of the well. Let's denote the depth of the well as Yw.
First, we need to find the time it takes for the sound to travel from the bottom of the well to the top.
Using the formula: distance = speed × time
Distance traveled by sound = Ys - Yw
Speed of sound = 344 m/s
Time taken by sound = (Ys - Yw) / 344
Next, we need to find the time it takes for the rock to reach the bottom of the well.
Using the formula for free fall: distance = 0.5 × acceleration × time^2
Distance traveled by the rock = Yr - Yir (initial velocity is zero for a dropped rock)
Acceleration due to gravity = -9.8 m/s^2 (assuming standard Earth gravity)
Time taken by rock = sqrt((2 × (Yr - Yir)) / -9.8)
Since the time it takes for the sound to travel up is 3 seconds, we can set up the equation:
(Ys - Yw) / 344 = 3 - sqrt((2 × (Yr - Yir)) / -9.8)
Simplifying and rearranging the equation, we can solve for Yw (the depth of the well):
Yw = Ys - 344(3 - sqrt((2 × (Yr - Yir)) / -9.8))
B) What are the final positions of the Rock and the Sound? (Rock = Yfr; Sound = Yfs)
To find the final positions of the rock and the sound, we need to calculate their displacements from their initial positions.
Displacement of the rock (Yfr) is given by:
Yfr = Yir + 0.5 * acceleration * time^2
where acceleration is -9.8 m/s^2 and the time taken is the same as calculated above.
Displacement of the sound (Yfs) is given by:
Yfs = Ys + speed * time
where speed is 344 m/s and the time taken is 3 seconds.
Substituting the values into the equations, you can find the final positions of the rock and the sound.