Suppose the slope of a beach underneath the ocean is 21 cm of dropoff for every 1.2 m of horizontal distance. A wave is moving inland, slowing down as it enters shallower water. What is its acceleration when it is 16 m from the shoreline? (Let the +x direction be towards the shoreline. Indicate the direction with the sign of your answer.)
To find the acceleration of the wave, we need to determine its change in velocity over time. Given the slope of the beach and the distance from the shoreline, we can calculate the change in velocity and divide it by the time taken for the wave to travel that distance.
1. First, we need to find the change in velocity. The change in velocity is equal to the change in vertical speed because the wave is moving parallel to the shoreline.
2. To calculate the change in vertical speed, we need to find the change in height of the beach (dropoff) when the wave travels the horizontal distance of 16 m.
3. The slope of the beach is given as 21 cm of dropoff for every 1.2 m of horizontal distance. We can calculate the dropoff of the beach over 16 m by setting up a proportion:
(21 cm / 1.2 m) = (x cm / 16 m)
Solving for x, we have:
x = (21 cm / 1.2 m) * 16 m
x ≈ 280 cm
4. The dropoff is approximately 280 cm, which is the change in height. But to calculate the change in velocity, we need to convert the change in height to meters. 1 meter is equal to 100 cm, so we have 280 cm / 100 cm/m = 2.8 m.
5. Now that we know the change in height is 2.8 m, we can use the formula for motion under constant acceleration to find the acceleration. The formula is:
v^2 = u^2 + 2as,
where v is the final velocity, u is the initial velocity (assumed to be zero as the wave slows down), a is the acceleration, and s is the distance traveled.
Rearranging the formula, we have:
a = (v^2 - u^2) / 2s
Since the initial velocity u is zero, the formula simplifies to:
a = v^2 / 2s
6. Plugging in the values, we get:
a = (0 m/s)^2 / (2 * 16 m) = 0 m/s^2
Therefore, the acceleration of the wave is 0 m/s^2.
Note: The wave slows down as it enters shallower water, but at the specific point where it is 16 m from the shoreline, it does not experience any acceleration.