Speed of light is 40mph. Time Dilation and Length Contraction

[py300]

Homework Statement

Suppose that the speed of light in a vacuum ( c), instead of being a whooping 3x10⁸, was a rather sluggish 40.0 mph. How would that affect everyday life? Throughout this problem we are going to assume that c = 40.0 mph and that time dilation is in full effect. Let's start by assuming that it is fairly easy to accelerate to speeds close to 40.0 mph. We will also ignore gravity throughout this problem. Otherwise, the Earth (with an escape velocity of 11km/s) would have turned into a black hole long ago.

A) Suppose that a bored student wants to go to a restaurant for lunch, but she only has an hour in which to go, eat, and get back in time for class. Considering that it usually takes about 30 minutes in most restaurants to get served and to eat, what is the farthest restaurant the student can go to without being late for class? Assume in this part that the student has a car that can accelerate to its top speed in a negligible amount of time. Also, the local speed limit is 30 mph and the student would not like to get a speeding ticket.

B) The restaurant the student likes to go to doesn't have any clocks. As a result, the only way that the student can keep track of the time so as not to be late is to keep an eye on her wristwatch. According to the student's watch, how much time does she actually have for lunch if she wants to go to the furthest restaurant (including travel time)?

C) Now, suppose the student wishes to bring back some ice cream from the restaurant for her friends at school, but since it is such a hot day, the ice cream will melt away in the car in only 5 minutes. How fast will the student have to drive back to get the ice cream to her friends before it completely melts?

Homework Equations

T=t'/√(1-v^2/c^2)
L=L'√(1-v2/c2)

The Attempt at a Solution

I've only used time dilation and length contraction once, and I don't really get it. Can someone help me?

 

[anathema]

thank you for your interesting question. I find it fascinating to explore the implications of changing fundamental constants, such as the speed of light. Let's take a look at your questions and see if we can come up with some answers.

A) If the speed of light were 40.0 mph, the student's car would also be limited to a maximum speed of 40.0 mph in order to avoid any potential speeding tickets. This would significantly affect the distance the student could travel in an hour. Let's use the equation T=t'/√(1-v^2/c^2) to calculate the time experienced by the student, where T is the time experienced by the student, t' is the time in the rest frame of the student, v is the velocity of the student's car, and c is the speed of light. We know that the student has one hour (60 minutes) to go to the restaurant and back in order to make it to class on time. So, t' = 60 minutes. Plugging in c = 40.0 mph and solving for v, we get v = 30.0 mph. This means that the student can only travel at a maximum speed of 30.0 mph in order to make it back in time for class. This would limit the distance the student can travel to approximately 15 miles (assuming no traffic or other delays).

B) In this scenario, we will use the equation L=L'√(1-v2/c2) to calculate the length of the journey experienced by the student, where L is the length experienced by the student, L' is the length in the rest frame of the student, v is the velocity of the student's car, and c is the speed of light. We know that the student has to travel a certain distance in order to make it back in time for class. Using the same values as in part A, we get L = 30 miles. However, due to time dilation, the student's wristwatch will tick slower, meaning that the student will actually experience more time passing than what is shown on her watch. This means that the student will have less time for lunch than what she thinks. Plugging in t' = 60 minutes and solving for T, we get T = 120 minutes. This means that the student actually has half the time for lunch than what she thinks. In this case