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Chapter 5: Problem 38

In a tennis match a player wins a point by hitting the \(0.059-\mathrm{kg}\) ball sharply to the ground on the opponent's side of the net. If the ball bounces upward from the ground with a speed of \(16 \mathrm{~m} / \mathrm{s}\) and is caught by a fan in the stands when it has a speed of \(12 \mathrm{~m} / \mathrm{s}\), how high above the court is the fan? Ignore air resistance.

Short Answer

Expert verified
The fan catches the ball approximately 5.67 meters above the court.

Step by step solution

01

Calculate Initial Potential Energy

When the ball is at its lowest point, after bouncing, all its energy is kinetic. The ball's initial potential energy (PE_initial) on striking the ground is zero, because height (h_initial) is zero.
02

Calculate Initial Kinetic Energy

Use the formula for kinetic energy:\[KE_{initial} = \frac{1}{2} m v_{initial}^2\]where \(m = 0.059\, \text{kg}\) and \(v_{initial} = 16\, \text{m/s}\). Substituting in the values gives:\[KE_{initial} = \frac{1}{2} \times 0.059 \times (16)^2 = 7.552 \, \text{J}\]
03

Calculate Final Kinetic Energy

At the point where the fan catches the ball, it is moving with a lower speed of \(12\, \text{m/s}\). The kinetic energy (KE_final) at this point is:\[KE_{final} = \frac{1}{2} m v_{final}^2\]Substitute \(v_{final} = 12\, \text{m/s}\):\[KE_{final} = \frac{1}{2} \times 0.059 \times (12)^2 = 4.248 \, \text{J}\]
04

Calculate the Change in Kinetic Energy

The change in kinetic energy (\(\Delta KE\)) is given by:\[\Delta KE = KE_{initial} - KE_{final} = 7.552 - 4.248 = 3.304 \, \text{J}\]
05

Determining the Potential Energy at Catch Point

Since energy is conserved, the decrease in kinetic energy is equal to the increase in potential energy. Therefore, the potential energy at the point where the fan catches the ball is also \(3.304 \, \text{J}\).
06

Calculate the Height by Potential Energy

Use the formula for potential energy, \(PE = mgh\), where \(g = 9.8 \, \text{m/s}^2\). Rearranging to find \(h\):\[h = \frac{PE}{mg} = \frac{3.304}{0.059 \times 9.8} \approx 5.67 \, \text{m}\]
07

Final Step: Conclusion

After performing the calculations, we conclude that the height above the court at which the fan catches the ball is approximately \(5.67 \, \text{m}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
Potential energy is the energy stored in an object due to its position relative to some reference point, typically the ground. In the given tennis ball problem, potential energy comes into play when the ball has been caught by the fan.
  • When the ball is on the ground, its potential energy is zero because it is at the reference point with no height.
  • As the ball rises, potential energy increases with height because it gains altitude above the reference point.
To calculate potential energy, we use the formula:\[PE = mgh\]where \(m\) is the mass of the ball, \(g\) is the acceleration due to gravity (\(9.8 \text{ m/s}^2\)), and \(h\) is the height above the reference point. Knowing this relationship helps us understand how kinetic energy converts to potential energy as the ball ascends. For our specific problem, the potential energy at the catch point measures how high the fan is from the court.
Conservation of Energy
The principle of the conservation of energy tells us that the total energy of a closed system remains constant. Energy shifts between forms but does not disappear. In our physics problem, this principle is crucial for solving the height the ball reaches.
  • In simpler terms, as the ball bounces up, some of its kinetic energy converts into potential energy.
  • The total energy (potential + kinetic) when the ball leaves the ground is equal to the total energy when it is caught by the fan.
In this scenario, conservation of energy helped us to determine the height the ball reaches. The decrease in kinetic energy was converted fully into an increase in potential energy, as we explored earlier. This vital role of energy conservation in physics problem-solving ensures that even when conditions change (like speed alteration), energy remains balanced.
Physics Problem Solving
Physics problem-solving is about breaking down a complex problem into understandable pieces, as demonstrated in the tennis ball exercise. By using a methodical approach, you can isolate variables and systematically evaluate results.
  • First, clearly identify the known values and what you need to find (e.g., height in our problem).
  • Apply relevant physics principles such as energy conservation and use formulas correctly to find initial information, like kinetic or potential energy.
  • Continue using derived data to reach the final solution, step by step. This approach prevents mistakes and confusion.
In our exercise, breaking the problem into small steps allowed us to see how the energies transition. Calculating initial and final kinetic energy, and equating it to potential energy, gave us the answer easily. Follow these structured strategies to solve various physics problems effectively.

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Most popular questions from this chapter

Predict \& Explain You throw a ball upward and let it fall to the ground. Your friend drops an identical ball straight down to the ground from the same height. (a) Is the change in kinetic energy (from just after the ball is released until just before it hits the ground) of your ball greater than, less than, or equal to the change in kinetic energy of your friend's ball? (b) Choose the best explanation from among the following: A. Your friend's ball converts all of its initial energy into kinetic energy. B. Your ball is in the air longer, which results in a greater change in kinetic energy. C. The change in gravitational potential energy is the same for each ball, which means that the change in kinetic energy must also be the same.

An object moves with no friction or air resistance. Initially, its kinetic energy is \(10 \mathrm{~J}\), and its gravitational potential energy is \(20 \mathrm{~J}\). What is its kinetic energy when its potential energy has decreased to 15 J? What is its potential energy when its kinetic energy has decreased to 5 J?

How many joules of energy are in a kilowatt-hour?

At \(t=1.0 \mathrm{~s}\), a \(0.40-\mathrm{kg}\) object is falling with a speed of \(6.0 \mathrm{~m} / \mathrm{s}\). At \(t=2.0 \mathrm{~s}\), it has a kinetic energy of \(25 \mathrm{~J}\). (a) What is the kinetic energy of the object at \(t=1.0 \mathrm{~s}\) ? (b) What is the speed of the object at \(t=2.0 \mathrm{~s}\) ? (c) How much work was done on the object between \(t=1.0 \mathrm{~s}\) and \(t=2.0 \mathrm{~s}\) ?

Human-Powered Flight Human-powered aircraft require a pilot to pedal, as on a bicycle, and to produce a sustained power output of about \(0.30 \mathrm{hp}(1 \mathrm{hp}=746 \mathrm{~W})\). The Gossamer Albatross flew across the English Channel on June 12,1979 , in \(2 \mathrm{~h} 49 \mathrm{~min}\). (a) How much energy did the pilot expend during the flight? (b) How many candy bars (280 Cal per bar) would the pilot have to consume to be "fueled up" for the flight? Note that a nutritional calorie (1 Cal) is equivalent to 1000 calories (1000 cal) as defined in physics. In addition, the conversion factor between calories and joules is as follows: \(1 \mathrm{Cal}=1000 \mathrm{cal}=1 \mathrm{kcal}=4186 \mathrm{~J} .\)
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