28.8

Concepts & Calculations

There are many astonishing consequences of special relativity, two of which are time dilation and length contraction. Example 12 reviews these important concepts in the context of a golf game in a world where the speed of light is a little faster than that of a golf cart.

| Concepts & Calculations Example 12 | | | Golf and Special Relativity

Imagine playing golf in a world where the speed of light is only c = 3.40 m/s. Golfer A drives a ball down a flat horizontal fairway for a distance that he measures as 75.0 m. Golfer B, riding in a cart, happens to pass by just as the ball is hit (see Figure 28-15). Golfer A stands at the tee and watches while golfer B moves down the fairway toward the ball at a constant speed of 2.80 m/s. (a) How far is the ball hit according to a measurement made by golfer B? (b) According to each golfer, how much time does it take for golfer B to reach the ball?

Figure 28-15 Just as golfer A hits the ball, golfer B passes by in a golf cart. According to special relativity, each golfer measures a different distance for how far the ball is hit.

Concept Questions and Answers Who measures the proper length of the drive, and who measures the contracted length?

Answer Consider two coordinate systems, one attached to the earth and the other to the golf cart. The proper length L0 is the distance between two points as measured by an observer who is at rest with respect to them. Since golfer A is standing at the tee and is at rest relative to the earth, golfer A measures the proper length of the drive, which is L0 = 75.0 m. Golfer B is not at rest with respect to the two points, however, and measures a contracted length for the drive that is less than 75.0 m.

Who measures the proper time interval, and who measures the dilated time interval?

Answer The proper time interval is the time interval measured by an observer who is at rest in his or her coordinate system and who views the beginning and ending events as occurring at the same place. The beginning event is when the ball is hit, and the ending event is when golfer B arrives at the ball. When the ball is hit, it is alongside the origin of golfer B’s coordinate system. When golfer B arrives at the ball down the fairway, it is again alongside the origin of his coordinate system. Thus, golfer B measures the proper time interval. Golfer A, who does not see the beginning and ending events occurring at the same place, measures a dilated, or longer, time interval.

Solution

a. | Golfer A, being at rest with respect to the beginning and ending points, measures the proper length of the drive, so L0 = 75.0 m. Golfer B, who is moving, measures a contracted length L given by Equation 28.2:

Thus, the moving golfer measures the length of the drive to be a shortened 42.5 m rather than the 75.0 m measured by the stationary golfer.
b. | According to golfer B, the time interval it takes to reach the ball is equal to the contracted length L that he measures divided by the speed of the ground with respect to him. The speed of the ground with respect to golfer B is the same as the speed v of the cart with respect to the ground. Thus, we find

Golfer A, standing at the tee, measures a dilated time interval , which is related to the proper time interval by Equation 28.1:

In summary, golfer A measures the proper length (75.0 m) and a dilated time interval (26.8 s), and golfer B measures a shortened length (42.5 m) and a proper time interval (15.2 s).

Other important consequences of special relativity are the equivalence of mass and energy, and how kinetic energy depends on the total and rest energies. Example 13 compares these properties for three different particles.

| Concepts & Calculations Example 13 | | | Mass and Energy

The rest energy E0 and the total energy E of three particles, expressed in terms of a basic amount of energy , are listed in the table below. The speeds of these particles are large, in some cases approaching the speed of light. For each particle, determine its mass and kinetic energy.

Particle | Rest Energy | Total Energy
a |  | 2
b |  | 4
c | 5 | 6

Concept Questions and Answers Given the rest energies specified in the table, what is the ranking (largest first) of the masses of the particles?

Answer The rest energy is the energy that an object has when its speed is zero. According to special relativity, the rest energy E0 and the mass m are equivalent. The relation between the two is given by Equation 28.5 as E0 = mc2, where c is the speed of light in a vacuum. Thus, the rest energy is directly proportional to the mass. From the table it can be seen that particles a and b have identical rest energies, so they have identical masses. Particle c has the greatest rest energy, so it has the greatest mass. The ranking of the masses, largest first, is c, then a and b (a tie).

Is the kinetic energy KE given by the expression , and what is the ranking (largest first) of the kinetic energies of the particles?

Answer No. The expression applies only when the speed of the object is much, much less than the speed of light. According to special relativity, the kinetic energy is the difference between the total energy E and the rest energy E0, so KE = E − E0. Therefore, we can examine the table and determine the kinetic energy of each particle in terms of E′. The kinetic energies of particles a, b, and c are, respectively, , and . The ranking of the kinetic energies, largest first, is b, then a and c (a tie).

Solution

a. | The mass of particle a can be determined from its rest energy . Since (see the table), its mass is

In a similar manner, we find that the masses of particles b and c are

As expected, the ranking is .
b. | According to Equation 28.6, the kinetic energy KE of a particle is equal to its total energy E minus its rest energy E0; . For particle a, its total energy is and its rest energy is , so its kinetic energy is

The kinetic energies of particles b and c can be determined in a similar fashion:

As anticipated, the ranking is .

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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