Problem 57 - Fundamentals of Physics

Question

To test the quality of a tennis ball, you drop it onto the floor from a height of \(4.00 \mathrm{~m}\). It rebounds to a height of \(2.00 \mathrm{~m}\). If the ball is in contact with the floor for \(12.0 \mathrm{~ms}\), (a) what is the magnitude of its average acccleration during that contact and (b) is the average acceleration up or down?

Step-by-Step Solution

Answer

The average acceleration is 1259.17 m/s² upwards.

Step 1: Find the Initial Velocity Before Impact

Using energy conservation, find the velocity just before the ball hits the floor. The initial potential energy is converted into kinetic energy.\[ mgh = \frac{1}{2}mv^2 \]Given, initial height \( h = 4.00 \, \text{m} \) and final velocity \( v \) we \[ v = \sqrt{2gh} \]where \( g = 9.81 \, \text{m/s}^2 \), so:\[ v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 4.00 \, \text{m}} = 8.85 \, \text{m/s} \] The velocity just before impact is \( -8.85 \, \text{m/s} \) (negative because it is directed downward).

Step 2: Find the Final Velocity After Rebounding

Now, calculate the velocity of the ball as it rebounds to a height of 2.00 m using the same energy principle as in step 1, in reverse.\[ v = \sqrt{2gh} \]Here, \( h = 2.00 \, \text{m} \), therefore:\[ v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 2.00 \, \text{m}} = 6.26 \, \text{m/s} \] The velocity just after impact is \( 6.26 \, \text{m/s} \) (positive because it is directed upward).

Step 3: Calculate Change in Velocity

The change in velocity (\( \Delta v \)) during the contact with the floor can be found by subtracting the initial velocity before impact from the final velocity after rebound:\[ \Delta v = v_{\text{after}} - v_{\text{before}} = 6.26 \, \text{m/s} - (-8.85 \, \text{m/s}) = 15.11 \, \text{m/s} \]

Step 4: Compute Average Acceleration

The average acceleration ( \( a \)) can be calculated using the formula:\[ a = \frac{\Delta v}{\Delta t} \]where \( \Delta t = 12.0 \, \text{ms} = 0.012 \, \text{s} \)\[ a = \frac{15.11 \, \text{m/s}}{0.012 \, \text{s}} = 1259.17 \, \text{m/s}^2 \] The magnitude of the average acceleration is \( 1259.17 \, \text{m/s}^2 \).

Step 5: Determine Direction of Average Acceleration

Since the average acceleration is positive, it means the direction is upwards, opposite the initial velocity direction.

Key Concepts

Energy Conservation

Energy conservation plays a crucial role in understanding the behavior of objects in motion, such as a bouncing tennis ball. When the tennis ball is released from a height, it possesses potential energy due to its position above the ground. As the ball falls, this potential energy is converted into kinetic energy.
The concept of energy conservation states that in a closed system, energy cannot be created or destroyed—it is merely converted from one form to another. In the case of the tennis ball, the total mechanical energy (potential + kinetic) remains constant during its fall and after it rebounds, ignoring air resistance and energy lost during contact with the floor.

• Potential energy at height: \[ ext{PE} = mgh \]-
• Kinetic energy as it hits the floor: \[ ext{KE} = rac{1}{2}mv^2 \]

Understanding these energy conversions allows us to determine the speed at which the ball impacts and rebounds effectively.

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. For the tennis ball in our exercise, kinetic energy is crucial at two points: when the ball is just about to hit the floor, and as it leaves the floor to rebound.
When the ball hits the floor, its speed is at a maximum and its kinetic energy is also at a peak. The formula for kinetic energy is:\[ ext{KE} = rac{1}{2} mv^2 \]Where

• \( m \) is the mass of the tennis ball, and
• \( v \) is the velocity.

By knowing the kinetic energy, one can easily determine the velocity of the ball at impact, which is essential for further calculations like average acceleration and energy loss during the bounce.

Potential Energy

Potential energy describes the stored energy in an object due to its position relative to other objects. It is heavily influenced by height when discussing gravitational potential energy. For the tennis ball problem, the potential energy converts largely to kinetic energy as the ball begins its free fall.
The formula for gravitational potential energy is:\[ ext{PE} = mgh \]Where

• \( m \) is the mass,
• \( g \) is the acceleration due to gravity \(9.81 ext{m/s}^2\), and
• \( h \) is the height above the ground.

The potential energy is maximum at the top where the tennis ball begins its fall and decreases as the ball descends.

Velocity Calculation

Calculating a tennis ball's velocity during free fall or rebound involves using the principles of energy transformation. As energy conservation applies, the potential energy transforms into kinetic energy during these motions. The calculation of velocity is central to many physics problems, including this one.
Velocity just before impact or after rebounding is determined using the modified equation derived from energy conservation:\[ v = \sqrt{2gh} \]This equation allows us to calculate the velocity from heights just before impact and right after rebound. The velocities were found to be \(-8.85 \, \text{m/s}\) before hitting and \(6.26 \, \text{m/s}\) after rebounding, emphasizing the concept of direction in velocity. The negative indicates the ball's descent, and positive indicates upward rebounding.