Einstein and Lorentz, being avid tennis players, play a...

Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 m from each other. Being very skilled players, they play without a net. The tennis ball has mass 0.0580 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 m/s. What is the ball's kinetic energy? (b) Einstein slams a return at 1.80 $\times$ 10$^8$ m/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 2.20 $\times$ 10$^8$ m/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 m, according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

Key Concepts

Time Dilation

Time dilation refers to the effect whereby time intervals measured in a moving frame appear longer compared to those measured in the rest frame. This means that clocks in motion relative to an observer are seen to tick slower by a factor of ?, illustrating how time is affected by relative motion in high-speed scenarios.

Length Contraction

Length contraction is the phenomenon in which the length of an object moving relative to an observer is measured to be shorter than its proper length. This contraction occurs only in the direction of the relative motion and is quantified by the factor 1/?, highlighting the non-absolute nature of spatial measurements in special relativity.

Relativistic Kinetic Energy

This concept involves calculating the kinetic energy of objects moving at speeds that are significant fractions of the speed of light, where classical formulas no longer apply. Instead of using the classical 1/2 mv² formula, the relativistic kinetic energy is given by (? - 1)mc², where ? (the Lorentz factor) accounts for the increased inertia of objects at high speeds.

Lorentz Factor

The Lorentz factor, denoted by ? = 1/?(1 - v²/c²), is central to the calculations in special relativity. It quantifies the effects of time dilation, length contraction, and the increase in relativistic mass, making it essential for converting quantities measured in one inertial frame to those in another moving at a relative velocity.

Relativistic Velocity Addition

At speeds near the speed of light, velocities do not simply add together as they do in classical mechanics. The relativistic velocity addition formula, u' = (u + v)/(1 + uv/c²), ensures that the resultant velocity never exceeds the speed of light and correctly accounts for the geometric structure of spacetime in special relativity.

Transcript

00:01 This is chapter 37 problem number 60.

00:03 Lawrence and einstein are playing tennis.

00:05 They're 20 meters away from each other.

00:07 The mass of the ball is given to us as 0 .058 kilograms.

00:13 And lawrence serves the ball at 80 meters per second.

00:17 And in part a, we're asked to calculate the kinetic energy of the ball if the speed of the ball is 80 meters per second.

00:25 So this is a non -relativistic regime done because you're dealing with speeds much lower than the speed of light, then the kinetic energy is going to be one -half m .b squared, right? and one -half and being the mass of the ball of 5, 8 kilograms, and the speed of the ball is going to be 80 meters per second.

00:48 Of course, we're going to square it.

00:49 So then kinetic energy is going to be 185 .6 joules.

00:53 Now, in part b, einstein serves the ball back, that strikes the ball back towards lawrence in a very much greater speed of 1 .8 times 10 to per of 8 meters per second.

01:15 So now 1 .8 times 10 to power of 8 meters per second.

01:20 Now we're in the road service to regime and we're asked to calculate kinetic energy in this case.

01:26 In this case, we can't use half mb squared anymore because this v is in.

01:31 The order of the speed of light, we are going to have to remember the relativistic kinetic energy equation, gamma factor minus 1, the rest mass, times c squared.

01:43 Now, let's calculate what gamma is here.

01:46 In this case, gamma general formula 4 is 1 minus v squared over c squared.

01:52 In our case, 1 over square of 1, minus 1 .8 times 10, or 8, 8 per second.

02:01 Squared divided by square of the speed of light.

02:07 So if you do the algebra, what we have here is then 1 .25 for our gamma factor.

02:14 Now, we're putting this back in the formula here, the kinetic energy formula here, 1 .25 minus 1 times the rest mass of the ball is 0 .058 kilograms.

02:29 C squared will be three times center per eight meters per second squared.

02:35 Then kinetic energy is going to be 1 .3 all 5 times center of seven duels.

02:44 Now in, let's start part c on a separate page.

02:50 In part c, we're given the, now we have a rabbit that is running from einstein towards lawrence, is their point in tennis.

03:01 So rabbit speed relative to those men is, i'm going to call it v prime, is 2 .2 times 10 to per 8 meters per second.

03:09 That's a quite fast rabbit.

03:12 Now, the bowl's speed was 1 .8 times 10 to per 8 meters per second...

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