Question

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

The fan is about 5.71 meters above the court.

Step-by-step solution

Calculate Initial Kinetic Energy

The kinetic energy (KE) when the ball bounces off the ground can be calculated using the formula: \[ \text{KE}_1 = \frac{1}{2} m v_1^2 \]where \( m = 0.059 \text{ kg} \) is the mass and \( v_1 = 16 \text{ m/s} \) is the initial speed.Plugging in the values, we get:\[ \text{KE}_1 = \frac{1}{2} \times 0.059 \times 16^2 = 7.552 \text{ J} \]

Calculate Kinetic Energy at the Fan's position

The kinetic energy (KE) of the ball when caught by the fan can be calculated with the same formula: \[ \text{KE}_2 = \frac{1}{2} m v_2^2 \]where \( v_2 = 12 \text{ m/s} \) is the speed when caught.Plugging in the values, we get:\[ \text{KE}_2 = \frac{1}{2} \times 0.059 \times 12^2 = 4.248 \text{ J} \]

Calculate Potential Energy Gain

The difference in kinetic energy as the ball rises is converted into potential energy (PE):\[ \Delta \text{KE} = \text{KE}_1 - \text{KE}_2 = 7.552 \text{ J} - 4.248 \text{ J} = 3.304 \text{ J} \]This difference represents the gain in gravitational potential energy \( \text{PE}_g \):\[ \text{PE}_g = mgh \]

Solve for Height

To find the height \( h \), rearrange the potential energy formula:\[ h = \frac{\Delta \text{KE}}{mg} = \frac{3.304}{0.059 \times 9.81} \]Calculate the height:\[ h \approx \frac{3.304}{0.57879} \approx 5.71 \text{ meters} \]

Conclude the Solution

The fan is approximately 5.71 meters above the court. This height ensures that the change in velocity from 16 m/s to 12 m/s can account for the increase in potential energy corresponding to this height gain.

Key concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It can be calculated using the formula: \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. In our tennis ball example, the ball's kinetic energy changes as it is hit from the ground upwards and then caught by a fan. Initially, when the tennis ball bounces with a speed of 16 m/s, it has a kinetic energy of approximately 7.552 Joules. This energy is a result of the ball's motion from the ground upwards. As it reaches the fan with a speed of 12 m/s, its kinetic energy is reduced to about 4.248 Joules. This decrease occurs because some kinetic energy is transformed into potential energy as the ball gains height.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. The higher an object is, the more potential energy it has. This energy can be expressed as \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)), and \( h \) is the height above a reference point. In our scenario, as the tennis ball rises toward the fan, it gains potential energy while losing kinetic energy. The gain in potential energy is equal to the decrease in kinetic energy, which is calculated to be 3.304 Joules. This suggests the ball rises to a height where it acquires this much potential energy.

Conservation of Energy

The principle of conservation of energy states that the total energy of an isolated system remains constant. Energy can change forms, such as from kinetic to potential, but the total quantity remains the same. In the exercise, we apply this principle by observing that as the ball rises, its kinetic energy decreases while its potential energy increases, maintaining a constant total energy. The initial total energy of 7.552 Joules when the ball leaves the ground equals the sum of its final kinetic energy and increased potential energy when caught by the fan. This balance shows how energy conservation allows us to solve for unknowns, such as the height to which the ball rises.

Physics Calculations

Physics calculations help us quantify the changes and interactions of physical systems using mathematical tools and formulae. They require identifying the right physical principles and equations. In this problem, we rely on kinetic and potential energy formulas to understand the motion and transfer of energy in the bouncing tennis ball.

• Calculate initial and final kinetic energies with \( KE = \frac{1}{2} mv^2 \).
• Determine potential energy gain using \( PE = mgh \).
• Use the conservation of energy principle \( KE_1 - KE_2 = mgh \) to find the change in height.

These calculations demonstrate practical applications of physics concepts in solving real-world problems like determining how high a ball goes based on its speed changes.