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 need to use the equations of motion. Let's break down each part of the problem:
(a) How deep is the lake?
To find the depth of the lake, we need to consider the ball's vertical motion. We can use the equation:
s = ut + (1/2)gt^2
Here, s represents the displacement, u is the initial velocity, t is the time, and g is the acceleration due to gravity.
In the first part of the problem, the ball is dropped from rest, so the initial velocity (u) is 0. The time (t) is given as 4.29 s, and we'll assume the acceleration due to gravity (g) is approximately 9.8 m/s^2.
By substituting the values, the equation becomes:
s = 0 * 4.29 + (1/2) * 9.8 * (4.29)^2
Calculating this equation will give us the depth of the lake.
(b) What is the magnitude of the average velocity of the ball for the entire fall?
To find the average velocity, we need to divide the total displacement by the total time taken. Since the ball starts from rest and ends with rest, the total displacement is 0. The total time taken is given as 4.29 s. So the average velocity will also be 0 m/s.
(c) What is the magnitude of the initial velocity of the ball when it is thrown from the diving board?
In this case, the ball reaches the bottom in the same time (4.29 s) as before, but it is now thrown from the diving board. We need to find the initial velocity (u) of the ball.
We can use the following equation (derived from the motion equation):
s = ut + (1/2)gt^2
Since the ball reaches the same depth as before, s will be the same value. We already know g and t. Now, we can solve the equation for u and find the magnitude of the initial velocity.
By using these steps, we can find the answers to all the parts of the problem.