A lead ball is dropped in a lake from a diving board 5.89 m above the water. It hits the water with a certain velocity and then sinks to the bottom with the same constant velocity. It reaches the bottom 4.29 s after it is dropped. (a) How deep is the lake? (b) What is the magnitude of the average velocity of the ball for the entire fall? (c) Suppose the water is drained from the lake. The ball is now thrown from the diving board so that it again reaches the bottom in 4.29 s. What is the magnitude of the initial velocity of the ball?
To solve this problem, we can use the principles of motion and the equations of motion to find the required information.
(a) Finding the depth of the lake:
In this case, we can assume the initial velocity of the ball when it is dropped is zero and the final velocity when it hits the bottom is also zero. We can use the equation of motion:
s = (u * t) + (0.5 * a * t^2)
Where:
s = displacement (depth of the lake) = unknown
u = initial velocity = 0
t = time taken to reach the bottom = 4.29 s
a = acceleration
Since the velocity remains constant after the ball hits the water, we can assume the acceleration to be zero during the time the ball is sinking. Thus, the equation becomes:
s = (u * t) + (0.5 * a * t^2)
s = 0 + (0.5 * 0 * 4.29^2)
s = 0
Therefore, the depth of the lake is 0 meters.
(b) Finding the magnitude of the average velocity of the ball:
Average velocity (v_avg) is given by the formula:
v_avg = total displacement / total time
Since the ball starts from rest and ends at rest, the total displacement is 0. The total time to reach the bottom is 4.29 s. Therefore, the magnitude of the average velocity is:
v_avg = 0 / 4.29
v_avg = 0 m/s
Therefore, the magnitude of the average velocity of the ball for the entire fall is 0 m/s.
(c) Finding the magnitude of the initial velocity of the ball when the water is drained from the lake:
To find the magnitude of the initial velocity, we can use the same equation of motion:
s = (u * t) + (0.5 * a * t^2)
In this case, the depth of the lake is known to be 0 (since the water is drained), the time taken to reach the bottom is 4.29 s, and we want to find the magnitude of the initial velocity (u). The equation becomes:
0 = (u * 4.29) + (0.5 * a * 4.29^2)
Since the final velocity is zero when the ball reaches the bottom, the term (0.5 * a * t^2) can be ignored. Simplifying the equation:
0 = u * 4.29
Solving for u:
u = 0
Therefore, the magnitude of the initial velocity of the ball when the water is drained from the lake is 0 m/s.