A lead ball is dropped into a lake from a diving board 5.24 m above the water. It hits the water with a certain velocity and then sinks to the bottom with this same constant velocity. It reaches the bottom 4.86 s after it is dropped. (Assume the positive direction is upward.)
(a) How deep is the lake?
(b) What is the average velocity of the ball?
(c) Suppose that all the water is drained from the lake. The ball is now thrown from the diving board so that it again reaches the bottom in 4.86 s. What is the initial velocity of the ball?
On the first, calculate the time it takes for something to drop 5.24m, then subtract that time from the total time. depth lake=vf*time
You can calculate vf from vf^2=2g h
and calculate time to surface..5.24=1/2gt^2
(a) To find the depth of the lake, we can use the formula for the distance traveled by an object in free fall:
d = (1/2)gt^2
where d is the distance, g is the acceleration due to gravity (approximately 9.8 m/s²), and t is the time.
In this case, the ball takes 4.86 seconds to reach the bottom, so we can substitute these values into the formula:
d = (1/2)(9.8)(4.86)^2
Calculating this gives us:
d ≈ 116.03 m
Therefore, the depth of the lake is approximately 116.03 meters.
(b) The average velocity of the ball can be found using the formula:
v_avg = d / t
where v_avg is the average velocity, d is the distance traveled, and t is the time.
Using the values we have, we can substitute them into the formula:
v_avg = 116.03 / 4.86
Calculating this gives us:
v_avg ≈ 23.86 m/s
Therefore, the average velocity of the ball is approximately 23.86 m/s.
(c) To find the initial velocity of the ball when it is thrown and reaches the bottom in 4.86 seconds, we can use the equation of motion:
d = v_i * t + (1/2)gt^2
In this case, the distance traveled is the same as the depth of the lake (116.03 m), the time is 4.86 seconds, and the acceleration due to gravity is 9.8 m/s².
Since the ball starts from rest when it is thrown, the initial velocity (v_i) will be the unknown. We can rearrange the equation to solve for v_i:
d = v_i * t + (1/2)gt^2
116.03 = v_i * 4.86 + (1/2)(9.8)(4.86)^2
Solving this equation will give us the initial velocity of the ball.