Question

Suppose the slope of a beach underneath the ocean is 20 cm of dropoff for every 1.6 m of horizontal distance. A wave is moving inland, slowing down as it enters shallower water. What is its acceleration when it is 12 m from the shoreline? (Let the +x direction be towards the shoreline. Indicate the direction with the sign of your answer.)

The answer is supposed to be -0.613m/s^2. I'm stumped on how to get this answer. I figured out that the total distance downward the beach goes at 12 m to be 1.5 m. However, I'm not sure if that's useful or not. Any help is appreciated.

Answer 1

The speed of a shallow wave is
v = sqrt(g h) where h is the water depth. The acceleration is thus:

a = dv/dt = 1/2 sqrt(g/h) dh/dt =

1/2 sqrt(g/h) dh/dt =

1/2 sqrt(g/h) dh/dx dx/dt =

1/2 sqrt(g/h) dh/dx v =

g/2 dh/dx =

9.81/2 m/s^2 (-0.2 m/1.6 m) =

-0.613 m/s^2