A lead ball is dropped in a lake from a diving board 6.28 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 5.16 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 5.16 s. What is the magnitude of the initial velocity of the ball?

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(a) To find the depth of the lake, we need to determine how far the ball falls while sinking to the bottom. The velocity of the ball while sinking is constant, and we can use the equation:

d = vt

Where d is the distance, v is the velocity, and t is the time taken. In this case, the time taken is 5.16 seconds, and we need to find the depth of the lake, so we'll call it d.

Since the ball is dropped from rest, the initial velocity is 0. The ball also falls for the same amount of time t = 5.16 s. Therefore, the distance fallen while sinking to the bottom is given by:

d = vt = (0 m/s) * (5.16 s) = 0 m

Therefore, the depth of the lake is 0 meters. This suggests that the situation described is inconsistent or incorrect, as the ball cannot sink in a lake with no depth.

(b) The magnitude of the average velocity can be determined by dividing the total distance traveled by the total time taken. Since the ball falls straight down, the total distance traveled is equal to the distance fallen from the diving board.

To calculate this distance, we can use the equation of motion:

d = (1/2) * g * t^2

where d is the distance, g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken.

Substituting the given values, we have:

d = (1/2) * (9.8 m/s^2) * (5.16 s)^2 = 126.6264 m

Therefore, the magnitude of the average velocity is the total distance divided by the total time:

v_avg = d / t = 126.6264 m / 5.16 s = 24.56 m/s

So, the magnitude of the average velocity of the ball for the entire fall is approximately 24.56 m/s.

(c) To find the magnitude of the initial velocity of the ball when it is thrown, we need to determine the distance it falls while sinking to the bottom. As mentioned in part (a), the distance fallen is 0 meters.

Since the distance fallen is 0, this implies that the initial velocity of the ball when it is thrown must be equal to the constant velocity it had while sinking to the bottom.

Therefore, the magnitude of the initial velocity of the ball is equal to the magnitude of the constant velocity while sinking, which is 24.56 m/s.