A lead ball is dropped into a lake from a diving board 5.42 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 4.98 s after it is dropped. (Assume the positive direction is upward.)
(a) How deep is the lake?
1Your answer is incorrect. m
(b) What is the average velocity of the ball?
2Your answer is incorrect. m/s
(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 4.98 s. What is the initial velocity of the ball?
Consider the free fall height of S=-5.42 m.
Initial velocity u=0 m/s.
Final velocity = v.
Acceleration due to gravity, g=-9.81 m/s/s
Use the relation
v²-u² = 2aS
to get
v=sqrt(2*(-9.81)*(-5.42)+0)
=10.312 m/s
The time required for the drop (to the surface of the lake)
= 10.312/9.81 m/s / m/s/s
= 1.051 seconds
There is enough information to complete the calculations. Post your work any time if you encounter problems.
To answer these questions, we can use the equations of motion and consider the motion of the ball in the vertical direction.
Let's start with question (a), which asks about the depth of the lake. We can use the equation of motion for free fall:
h = ut + (1/2)gt^2
Where:
h is the height or depth (unknown)
u is the initial velocity (unknown)
g is the acceleration due to gravity (-9.8 m/s^2)
t is the time taken (4.98 s)
Since the ball is dropped, the initial velocity u is zero. Substituting these values into the equation, we get:
h = 0 + (1/2)(-9.8)(4.98)^2
Simplifying the equation, we find:
h = -(1/2)(9.8)(4.98)^2
Evaluating this expression, we get:
h ≈ -122.5 m
Since the depth cannot be negative, we take the absolute value of the result to find that the lake is approximately 122.5 meters deep.
Now, let's move on to question (b), which asks about the average velocity of the ball. In this case, the average velocity can be found using the equation:
Average velocity = total displacement / total time
The total displacement is equal to the depth of the lake (which we found in question (a)) since the ball sinks to the bottom with a constant velocity. We already determined that the depth is approximately 122.5 m. The total time is given as 4.98 s.
So, the average velocity is:
Average velocity = 122.5 m / 4.98 s
Calculating it, we get:
Average velocity ≈ 24.6 m/s
Therefore, the average velocity of the ball is approximately 24.6 m/s.
Lastly, moving on to question (c), which asks about the initial velocity of the ball when it is thrown from the diving board. We can use the same equation of motion for free fall:
h = ut + (1/2)gt^2
In this case, the final height or depth h is still 122.5 m, time t is 4.98 s, and the acceleration due to gravity is still -9.8 m/s^2. However, the initial velocity u is unknown.
Rearranging the equation, we get:
u = (h - (1/2)gt^2) / t
Plugging in the values, we have:
u = (122.5 - (1/2)(-9.8)(4.98)^2) / 4.98
Evaluating this expression, we find:
u ≈ 122.5 m/s
Therefore, the initial velocity of the ball when it is thrown from the diving board is approximately 122.5 m/s.