Problem 96 - Fundamentals of Physics

Question

A lead ball is dropped in a lake from a diving board \(5.20 \mathrm{~m}\) ahove 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 \(4.80 \mathrm{~s}\) after it is dropped. (a) How deep is the lake? What are the (b) magnitude and (c) direction (up or down) of the average velocity of the ball for the entire fall? 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 \(4.80 \mathrm{~s}\). What are the (d) magnitude and (e) direction of the initial velocity of the ball?

Step-by-Step Solution

Answer

(a) 38.1 m deep, (b) 9.02 m/s, (c) down, (d) 14.6 m/s, (e) up.

Step 1: Determine the time to hit the water

The lead ball falls from a height of 5.20 m. We can use the kinematic equation for free fall to determine the time it takes to hit the water. The equation is \[s = ut + \frac{1}{2}gt^2\].Here, initial velocity \(u = 0\) (since it is dropped), \(s = 5.20\, \text{m}\), and acceleration due to gravity \(g = 9.81\, \text{m/s}^2\). Solving for time \(t\), we ignore the initial velocity term:\[5.20 = \frac{1}{2} \times 9.81 \times t^2\]\[t^2 = \frac{5.20 \times 2}{9.81}\]\[t^2 \approx 1.06\]\[t \approx 1.03\,\text{s}\].Therefore, it takes approximately 1.03 seconds to hit the water.

Step 2: Determine the velocity at water impact

Using the time from Step 1, use the kinematic equation to determine the velocity when the ball hits the water:\[v = u + gt\],where \(u = 0\), \(g = 9.81 \,\text{m/s}^2\), and \(t = 1.03 \,\text{s}\). \[v = 0 + 9.81 \times 1.03\]\[v \approx 10.1\,\text{m/s}\].The ball hits the water with a velocity of approximately 10.1 m/s.

Step 3: Calculate the depth of the lake

Since it reaches the bottom in a total of 4.80 seconds and it takes 1.03 seconds to hit the water, the time spent sinking is:\[T_{sink} = 4.80 - 1.03 = 3.77 \, \text{s}\,\].The sinking takes place at constant velocity, which is the velocity it had upon hitting the water. Therefore, the depth of the lake is calculated by:\[\text{Depth} = v \times T_{sink}\]\[\text{Depth} = 10.1 \times 3.77\]\[\text{Depth} \approx 38.1 \,\text{m}\].The lake is approximately 38.1 m deep.

Step 4: Magnitude of average velocity for entire fall

The average velocity \(v_{avg}\) for the entire fall is given by the formula:\[v_{avg} = \frac{\text{Total distance}}{\text{Total time}}\].The total distance is the height of the diving board plus the depth of the lake:\[\text{Total distance} = 5.20 + 38.1 = 43.3 \,\text{m}\].The total time is 4.80 s (given), so:\[v_{avg} = \frac{43.3}{4.80}\]\[v_{avg} \approx 9.02 \,\text{m/s}\].The magnitude of the average velocity for the entire fall is approximately 9.02 m/s.

Step 5: Direction of average velocity for entire fall

Since the ball is falling downwards throughout its journey from the board to the bottom, the direction of the average velocity is **down**.

Step 6: Initial velocity if thrown without water

The time to reach the bottom without water is still 4.80 seconds. We need to calculate the initial velocity \(u\) when the ball is thrown from the height (5.20 m plus depth 38.1 m = 43.3 m) using the kinematic equation:\[s = ut + \frac{1}{2}gt^2\].Here, \(s = 43.3\), \(t = 4.80\), \(g = 9.81\), solve for \(u:\)\[43.3 = u \times 4.80 + \frac{1}{2} \times 9.81 \times (4.80)^2\]\[43.3 = u \times 4.80 + 113.2\]\[u \times 4.80 = 43.3 - 113.2\]\[u \times 4.80 = -69.9\]\[u = \frac{-69.9}{4.80} \approx -14.6 \,\text{m/s}\].The initial velocity should be approximately 14.6 m/s upwards.

Step 7: Direction of initial velocity when thrown without water

Since we calculated the initial velocity as a negative value, it indicates that the direction is opposite to the gravitational force (i.e., upwards for the throw). Therefore, the initial direction when thrown is **up**.

Key Concepts

Free Fall

Free fall describes the motion of an object solely under the influence of gravity.
In this scenario, an object starts from rest, meaning it has an initial velocity of 0.
As the object falls, it accelerates due to gravity, typically at a rate of \(9.81 \, \text{m/s}^2\) on Earth.

• Free fall calculations often use the kinematic equation: \(s = ut + \frac{1}{2}gt^2\), where \(u\) is the initial velocity, \(s\) is the distance fallen, \(t\) is the time taken, and \(g\) is the acceleration due to gravity.
• In the given problem, the object was dropped from a diving board. The initial velocity (\(u\)) was zero.

After free falling for a certain time, it impacts the water with a measurable velocity.
This initial phase of motion relies solely on gravity.

Constant Velocity

Constant velocity implies a uniform motion where an object's speed and direction remain unchanged.
It doesn't speed up or slow down.
In the exercise, once the ball hits the water, it continues to sink at the same velocity it had upon impact.

• This concept can be referred to as terminal velocity in other scenarios, but in this exercise, it's simplified to the uniform motion underwater.
• The movement at constant velocity helps in calculating the depth of the lake by using the formula \(\text{Distance} = v \times t\).
• For the ball, this means it sinks uniformly at the speed it hits the water, simplifying underwater motion calculations.

This constant velocity is central for calculating the depth in which the ball traveled underwater.

Average Velocity

Average velocity is a measure of the total displacement of an object divided by the total time taken.
It encompasses all changes in velocity over the time interval and shows how fast something moves on average.

• The formula used is \(v_{avg} = \frac{\text{Total distance}}{\text{Total time}}\).
• In this problem, the average velocity takes into account both the fall and the sinking phase, totalling the distance from the diving board to the lake bottom.
• In terms of units, average velocity is often also measured in meters per second (m/s), which helps in understanding the overall motion in context.

It's a key concept to analyze both phases of motion synergistically, providing insight into overall movement.

Kinematic Equations

Kinematic equations are a set of formulas used to predict the future motion of an object, assuming constant acceleration (such as gravity).
They provide a powerful toolkit for solving various physics problems related to motion.

• Key equations include:
  • \(v = u + at\)
  • \(s = ut + \frac{1}{2}at^2\)
  • \(v^2 = u^2 + 2as\)
  These can be rearranged to solve for different variables like velocity (\(v\)), initial velocity (\(u\)), time (\(t\)), or displacement (\(s\)).
• In scenarios involving free fall, these equations guide predictions of how fast an object will move, its position after a certain time, and other dynamics related to motion.

Kinematics serve as the backbone for analyzing the problem, showing the importance of these equations in motion prediction.