How Deep Is the Lake and What Is the Initial Velocity of the Ball?

[lgen0290]

Homework Statement

A lead ball is dropped into a lake from a diving board 5.44 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 4.84 s after it is dropped. (Assume the positive direction is upward.)
(a) How deep is the lake?


(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 4.84 s. What is the initial velocity of the ball?

Homework Equations

h=.5(g)(t^2)

The Attempt at a Solution

I figured that the ball would fall at -9.8m/s, but that the trip it takes would have to be divided into two parts, as when it hit the surface of the water it would stop for an instant.

I figured 5.44m=.5(-9.8 m/s)(t^2). This gave me 1.05 s(which doesn't sound realistic, but the units cancel out find). I then took the remaining time(3.79 s) and plugged it into get
h=.5(-9.8 m/s)(3.79s^2) and got 70. 38 m.

For the other problem I figured there would be no stop, so I used h=.5(-9.8)(4.84^2) and got 114.74/4.84=23.708 m/s.

What gives?

 

[Tedjn]

For question (a),

You are correct that the time it takes for the ball to fall from the diving board to the water surface is 1.05 s (this does make sense, since with g = -9.8 m/s², a ball starting from rest would fall 4.9 m; 5.44 m is a little more than 4.9 m, so it takes a little longer than 1 s).

However, remember that the problem says that once the ball is inside the water, it travels with constant velocity. That means there is no more acceleration (presumably, the drag from the water will cancel out gravity). Therefore, you would not use -9.8 as your acceleration for the remaining 3.79 seconds.

For problem (c),

This problem depends on your answer to part (a). Currently, you are assuming the initial velocity is 0 and calculating h, which is how far the ball would travel. Actually, you know how far the ball has to travel. It is 5.44 m + the depth of the lake.

Instead, the question asks you what the initial velocity has to be for the ball to fall, always under the acceleration of gravity, g = -9.8 m/s², the distance 5.44 m + depth of lake in 4.84 s. You know the time of travel, the distance the ball must move, and the constant acceleration, and you are asked to find initial velocity. Which of your four kinematic equations should you use?

 

[Moetasim]

I would like to commend you for attempting to solve this problem using the appropriate equations and taking into account the different stages of the ball's motion. Your approach is correct, however, your calculations may have some errors.

Firstly, the initial velocity of the ball is not -9.8 m/s, as this is the acceleration due to gravity. The initial velocity is 0 m/s, as the ball is dropped from rest.

Secondly, in your first calculation, you used the time of 4.84 s, which is the total time for the ball to reach the bottom. However, in the second calculation, you used only the time of 3.79 s, which is the time for the ball to reach the bottom after hitting the water. This is why you got different results.

To solve for the depth of the lake, you can use the equation h = h1 + h2, where h1 is the distance the ball falls before hitting the water and h2 is the distance it falls after hitting the water. Using the equation h = 0.5gt^2, you can solve for h1 and h2 separately.

For the second part of the problem, you can use the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration (which is -9.8 m/s^2), and t is the time. You can solve for u by plugging in the given values for v, a, and t.

I hope this explanation helps you understand and solve the problem correctly. Keep up the good work!