One way to strictly enforce a speed limit would be to alter...

One way to strictly enforce a speed limit would be to alter the laws of nature. Suppose the speed of light were $65 \mathrm{mph}$ and your workplace was 30 miles from your home. Assume you travel to work at a typical driving speed of 60 mph. (a) If you drove at that speed for the round trip to and from work, light, how much would your wristwatch lag your kitchen clock each day? (b) Estimate the length of your car. (c) If you were driving at your estimated driving speed, how long would your car be when viewed from the roadside? (d) What would be the speed relative to you of similar cars traveling toward you in the opposite lane with the same ground speed as you? (e) How long would you measure those cars to be? (f) If the total mass of you and your car was $2000 \mathrm{~kg}$, how much work would be required to get you up to speed? (Note: Your rest mass energy in this world is $m c^{2}$, where $c=65$ mph. ) (g) How much work would be required in the real world, where the speed of light is $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s},$ to get you up to speed?

One way to strictly enforce a speed limit would be to alter the laws of nature. Suppose the speed of light were $65 \mathrm{mph}$ and your workplace was 30 miles from your home. Assume you travel to work at a typical driving speed of 60 mph. (a) If you drove at that speed for the round trip to and from work, light, how much would your wristwatch lag your kitchen clock each day? (b) Estimate the length of your car. (c) If you were driving at your estimated driving speed, how long would your car be when viewed from the roadside?
(d) What would be the speed relative to you of similar cars traveling toward you in the opposite lane with the same ground speed as you?
(e) How long would you measure those cars to be? (f) If the total mass of you and your car was $2000 \mathrm{~kg}$, how much work would be required to get you up to speed? (Note: Your rest mass energy in this world is $m c^{2}$, where $c=65$ mph. ) (g) How much work would be required in the real world, where the speed of light is $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s},$ to get you up to speed?

Key Concepts

Relativistic Kinetic Energy and Rest Mass Energy

In the realm of special relativity, the kinetic energy of an object is not given by the classical formula but instead must account for the relativistic increase in energy as speeds approach that of light. The concept of rest mass energy, expressed by the equation E = m c², indicates that even at rest, an object possesses intrinsic energy purely as a consequence of its mass. Calculations involving work or energy in high-speed contexts require these relativistic formulas to accurately capture the energy changes involved.

Relativistic Velocity Addition

This concept addresses how velocities combine when objects are moving at speeds that are significant fractions of the speed of light. Unlike classical addition of velocities, the formula for relativistic velocity addition ensures that the result never exceeds the speed of light, maintaining consistency with the core postulate of special relativity.

Special Relativity

This is the framework introduced by Einstein that describes the physics of objects moving at speeds close to that of light. It shows that the measurements of time, length, and mass are not absolute but depend on the relative motion between the observer and the object. The theory provides the reasoning behind non-intuitive phenomena such as time dilation, length contraction, and the need for a speed limit (the speed of light) in any consistent physical theory.

Time Dilation

Time dilation is the effect in relativity where a moving clock is observed to tick slower than a clock at rest in the observer's frame. This means that if one compares a clock traveling with an object moving at a high speed with a stationary clock, the moving clock will lag behind. This phenomenon plays a key role in scenarios where differences in elapsed times are calculated for journeys at speeds close to the speed of light.

Length Contraction

Length contraction is the relativistic concept that objects moving at a high speed will appear shorter along the direction of motion to a stationary observer. This effect does not mean the object physically shrinks; rather, space itself is measured differently in different inertial frames, leading to shorter observed lengths for objects in motion relative to an observer.

Transcript

00:01 For this problem on the topic of relativity, we want to suppose that the speed of flight was 65 miles an hour, your workplace was 30 miles from your home, and a typical driving speed would be 60 miles an hour.

00:12 Now, if you were to drive at that speed for a round trip to and from work, we want to know how by how much your wristwatch would lag your kitchen clock each day.

00:22 We then want to estimate the length of the car, and we want to know how long the car would be when viewed from the roadside.

00:30 We want to know then what would be the speed relative to you of similar cars traveling toward you in the opposite lane with the same ground speed as you and how long you would measure those cars to be.

00:42 If a total mass of you in the car was 2 ,000 kilograms, we want to know the work that would be required to get you up to that speed.

00:49 And lastly, how much work would be required in the real world where the speed of light is three times 10 to the 8 meters per second to get you up to speed.

00:58 Now we want the time difference between the kitchen clock and the moving wristwatch for the first part of problem a.

01:05 And so for the kitchen clock time, we drive at 60 miles, we drive 60 miles at 60 miles an hour.

01:14 And so the time would be one hour.

01:18 But measured by your wristwatch, the time is the proper time where delta t is gamma delta t -naut.

01:31 And gamma is equal to 1 over the square root of 1 minus u squared over c squared, which is 1 over the square root of 1 minus 60 over 65 squared, which gives 0 .0 .n's factor of 2 .6.

01:58 And so the time measured by the watch delta t -0 is delta t over gamma, which is one hour divided by 2 .6, which is 0 .38 of an hour, which is 23 minutes.

02:23 So the wristwatch would measure a time of 23 minutes.

02:26 Now for part b, the estimate of the length of the car would be 15 feet in length.

02:47 And for part c, we want its length l from a roadside observer, and this will be its proper length, l0, which is equal to 15 feet...

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