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.
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
To find the acceleration of the wave, you need to analyze the change in its velocity over time. This can be done by looking at its position as a function of time and differentiating twice with respect to time to find the acceleration.
Since the wave is moving slower in shallow water, it is decelerating, or experiencing negative acceleration. To determine the acceleration, we need to find the rate at which the wave's velocity is changing as it moves 12 m from the shoreline.
Let's break down the problem into smaller steps:
Step 1: Find the slope of the beach
You mentioned that the slope of the beach is 20 cm of dropoff for every 1.6 m of horizontal distance. This slope can be expressed as the ratio of the vertical change (dropoff) to the horizontal distance:
slope = vertical change / horizontal distance
slope = 20 cm / 1.6 m
Now, convert cm to meters:
slope = (20 cm / 100 cm) / 1.6 m
slope = 0.2 m / 1.6 m
Step 2: Calculate the vertical distance the wave drops when it moves 12 m horizontally
Using the slope you calculated in step 1, you can find the vertical distance the wave drops when it moves 12 m horizontally. This can be done by multiplying the horizontal distance by the slope:
vertical distance = slope * horizontal distance
vertical distance = 0.2 m / 1.6 m * 12 m
Step 3: Calculate the initial velocity of the wave at 12 m from the shoreline
Since the wave is slowing down as it enters shallower water, we need to calculate its initial velocity at 12 m from the shoreline. To do this, we can use the concept of conservation of energy, assuming that the wave's potential energy at the start (shallow water) is equal to its kinetic energy at 12 m from the shoreline.
Potential energy = Kinetic energy
mass * gravity * height = 0.5 * mass * (initial velocity)^2
mass and gravity cancel out:
height = 0.5 * (initial velocity)^2
Using the vertical distance calculated in step 2 as the height, we can solve for the initial velocity:
(initial velocity)^2 = 2 * height
(initial velocity)^2 = 2 * (vertical distance)
Step 4: Calculate the acceleration of the wave
Now that we have the initial velocity at 12 m from the shoreline, we need to find the time it takes for the wave to travel this distance. The wave's velocity will decrease uniformly over time, so we can use the kinematic equation:
distance = initial velocity * time + 0.5 * acceleration * time^2
Plugging in the known values:
12 m = initial velocity * time + 0.5 * acceleration * time^2
Since you want to find the acceleration, we need to solve this equation for acceleration. Rearranging the equation gives:
0.5 * acceleration * time^2 + initial velocity * time - 12 m = 0
Now, we can differentiate the position equation twice with respect to time to find the acceleration:
Differentiating once:
velocity = initial velocity + acceleration * time
Differentiating again:
acceleration = 0 + acceleration
In this case, the final value of acceleration is just acceleration itself, as there is no term involving time. So, we have the equation:
acceleration = acceleration
Given that the wave is decelerating, the acceleration will be negative. This explains why the answer is -0.613 m/s^2.
Note: It is worth mentioning that the method used here assumes a constant slope for the entire beach. In reality, beach slopes can vary, and wave behavior is more complex. This simplified approach provides a rough estimate based on the provided information.