How Deep Is The Well?

How Deep Is The Well?
TECTechnics Classroom TECTechnics Overview

A rascally boy drops a stone into a well and listens for the sound of the splash. He finds that 6.5 secs elapse from the instance the stone is dropped until he hears the sound of the splash. Assuming that sound travels at 1152 ft/sec, how far below is the surface of the water?

The string: S₇P₄A₄₁ (Linear Motion);
The Math:
How Deep Is The Well
Pj Problem of Interest (PPI) is of the type motion. Problems where distances traveled are sought are in general motion problems. In this case, it is the travel of the pebble that leads us to the depth of the well.
Let t₁ be the time it took for the stone to hit the surface of the water in the well.
Let t₂ be the time it took for the boy to hear the sound of the splash.
Then, t₁ + t₂ = 6.5 secs----(1).
Distance traveled by the stone = 16(t₁)²
Distance traveled by the sound from the splash = 1152t₂
Both distances measure d, the depth of the well to the surface of the water:
So, 16(t₁)² = 1152t₂ ----(2)
So, 16(t₁)² = 1152(13/2 - t₁)----(3)
because t₁ = 6.5 - t₂ (from equation 1).
Dividing equation (3) by 16, we have:
t₁² = 72(13/2 - t₁)
So, t₁² + 72t₁ - 468 = 0----(4)
Equation (4) is a quadratic equation
So, (t₁ + 78)(t₁ - 6) = 0
So, t₁ = 6 is the sensible solution of equation (4)
So, t₂ = 0.5 secs.
So, depth of well, d = 16(36) = 576 ft.
Also, d = 1152(0.5) = 576 ft (using t₂)

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