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

In summary, the problem involves a lead ball being dropped into a lake from a diving board and sinking to the bottom with constant velocity. The question asks for the depth of the lake and the initial velocity of the ball in two different scenarios. To solve these problems, you need to use the kinematic equations and take into account the different stages of the ball's motion.
  • #1
lgen0290
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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?
 
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  • #2
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?
 
  • #3


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!
 

What is free fall acceleration?

Free fall acceleration is the acceleration that an object experiences when it is falling under the sole influence of gravity. This acceleration is constant and is equal to 9.8 meters per second squared on Earth.

How is free fall acceleration calculated?

Free fall acceleration can be calculated using the formula a = g, where a is the acceleration and g is the gravitational acceleration, which is equal to 9.8 meters per second squared on Earth.

Does free fall acceleration depend on the mass of an object?

No, free fall acceleration does not depend on the mass of an object. This means that all objects, regardless of their mass, will experience the same acceleration when falling under the influence of gravity.

What is the difference between free fall and terminal velocity?

Free fall is the initial phase of falling when an object accelerates under the influence of gravity. Terminal velocity is the maximum velocity that an object can reach when falling due to the balancing of forces acting upon it.

How does air resistance affect free fall acceleration?

Air resistance, also known as drag force, can decrease the acceleration of an object in free fall. This is because the force of air resistance acts in the opposite direction of the object's motion, causing it to slow down. This is why objects with a larger surface area, like a feather, experience less acceleration due to air resistance compared to objects with a smaller surface area, like a rock.

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