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Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 \(\mathrm{m}\) from each other. Being very skilled players, they play with- out a net. The tennis ball has mass 0.0580 \(\mathrm{kg}\) . You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at 80.0 \(\mathrm{m} / \mathrm{s}\) . What is the ball's kinetic energy? (b) Einstein slams a return at \(1.80 \times 10^{8} \mathrm{m} / \mathrm{s}\) . What is the ball's kinetic energy? (c) During Einstein's return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of \(220 \times 10^{8} \mathrm{m} / \mathrm{s}\) relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run \(20.0 \mathrm{m},\) according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees a it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

Step by step solution

01

Calculate the Ball's Kinetic Energy during Lorentz's Serve

The kinetic energy (KE) of an object is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. For Lorentz's serve, \( m = 0.0580 \, \text{kg} \) and \( v = 80.0 \, \text{m/s} \). Plug these values into the formula:\[ KE = \frac{1}{2} \times 0.0580 \, \text{kg} \times (80.0 \, \text{m/s})^2 \]\[ KE = 0.029 \, \times 6400 \, \text{kg} \, \text{m}^2/\text{s}^2 = 185.6 \, \text{J} \].So, the ball's kinetic energy during Lorentz's serve is 185.6 J.

02

Calculate the Ball's Kinetic Energy during Einstein's Return

For Einstein's return, the velocity \( v \) is \( 1.80 \times 10^8 \, \text{m/s} \). The kinetic energy is calculated using the same formula:\[ KE = \frac{1}{2} \times 0.0580 \, \text{kg} \times (1.80 \times 10^8 \, \text{m/s})^2 \]\[ KE = 0.029 \, \times 3.24 \times 10^{16} \, \text{kg} \, \text{m}^2/\text{s}^2 \]\[ KE = 9.396 \times 10^{14} \, \text{J} \].Thus, the ball's kinetic energy during Einstein's return is \( 9.396 \times 10^{14} \, \text{J} \).

03

Determine the Rabbit's Speed Relative to the Ball

The rabbit runs beside the court at \( 220 \times 10^8 \, \text{m/s} \) relative to the men. The velocity of the ball is \( 1.80 \times 10^8 \, \text{m/s} \) in the same direction. The relative speed of the rabbit with respect to the ball is:\[ v_{\text{relative}} = v_{\text{rabbit}} - v_{\text{ball}} \]\[ v_{\text{relative}} = 220 \times 10^8 \, \text{m/s} - 1.80 \times 10^8 \, \text{m/s} \]\[ v_{\text{relative}} = 218.2 \times 10^8 \, \text{m/s} \].Thus, the speed of the rabbit relative to the ball is \( 218.2 \times 10^8 \, \text{m/s} \).

04

Calculate the Distance from Einstein to Lorentz Measured by the Rabbit

Due to relativistic effects, the distance the rabbit measures is affected by Lorentz contraction. The contracted length \( L \) is given by:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \( L_0 = 20.0 \, \text{m} \) is the proper length, \( v = 220 \times 10^8 \, \text{m/s} \), and \( c = 3 \times 10^8 \, \text{m/s} \).\[ L = 20.0 \, \text{m} \sqrt{1 - \left(\frac{220 \times 10^8 \, \text{m/s}}{3 \times 10^8 \, \text{m/s}}\right)^2} \].The value inside the square root will be negative, which implies a scenario faster than light and thus is not physically realizable under Newtonian physics. We have to consider classical assumptions have their limitations.

05

Calculate Time Taken by the Rabbit to Run 20.0 m

Using the formula \( t = \frac{d}{v} \), where \( d = 20.0 \, \text{m} \) and \( v = 220 \times 10^8 \, \text{m/s} \), the time \( t \) is calculated as:\[ t = \frac{20.0 \, \text{m}}{220 \times 10^8 \, \text{m/s}} \approx 0 \, \text{s} \].Since the rabbit's speed is unphysically high (beyond speed of light), this time calculation suggests an instantaneous traversal, assuming classical physics.

06

Time Measured by the Rabbit's Pocket Watch

As observed by the rabbit, time dilates according to \( \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \). \( \Delta t_0 \) is the proper time, and with \( v = 220 \times 10^8 \, \text{m/s} \), the relative Lorentz factor would lead to unphysical results under these assumed conditions. This illustrates that at relativistic speeds, classical interpretations require careful considerations.

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Key Concepts

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

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It's an important concept in both classical and relativistic physics, helping us understand how much work an object can perform or how much energy it carries in motion.

• The classical formula for kinetic energy is given by: \[ KE = \frac{1}{2} m v^2 \]Here, \( m \) is the mass, and \( v \) is the velocity of the object.
• In the scenario where Lorentz serves the tennis ball at 80.0 m/s, the kinetic energy is calculated as 185.6 J.
• However, because Einstein returns the ball at a speed close to the speed of light, \( 1.80 \times 10^8 \) m/s, the kinetic energy calculated using the classical formula seems extraordinarily high, reaching \( 9.396 \times 10^{14} \) J.

This disparity highlights the limitations of the classical kinetic energy formula when dealing with relativistic speeds. At such high speeds, relativistic effects must be considered to accurately describe the energy involved.

Lorentz Contraction

Lorentz contraction is a fascinating phenomenon that occurs when an object travels close to the speed of light. At such speeds, lengths appear contracted along the direction of motion.
When measuring the distance between two points from a moving frame:

• The contracted length \( L \) is given by the formula:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \( L_0 \) is the proper length (rest length), \( v \) is the object's velocity, and \( c \) is the speed of light.
• For the white rabbit chasing the tennis ball, this formula suggests the measured distance could contract when moving at relativistic speeds.

Interestingly, the calculations propose a negative number inside the square root because the rabbit鈥檚 speed exceeds the speed of light, hinting at scenarios beyond standard physics rules. This highlights that our understanding of physics is based on speed limits enforced by the speed of light.

Relativistic Speed

Relativistic speeds are those at which the effects of Einstein's theory of relativity become significant.

• As objects approach the speed of light \( c \), traditional physics starts to break down; special relativity gives us the tools to describe these conditions.
• For example, normal calculations suggested that the rabbit running at \( 220 \times 10^8 \) m/s would experience zero time passing to cover a distance鈥攁 paradox under classical physics.
• Time dilation and Lorentz contraction are among the primary effects seen at these speeds.

Overall, understanding relativistic physics deepens our appreciation for the universe's complexity and the limitations of classical perspectives when dealing with high velocities.