Homework 16

- Begin Date: 3/19/2025 12:01:00 AM
- Due Date: 3/26/2025 11:59:00 PM
- End Date: 3/28/2025 11:59:00 PM

Problem 5: (14% of Assignment Value)

Suppose that the speed of light in a vacuum ($c$), instead of being [tex]$3 \times 10^8 \, \text{m/s}$[/tex], was 40 mph. How would that affect everyday life? Assume [tex]$c = 40 \, \text{mph}$[/tex] and that time dilation is in full effect. Let's start by assuming it is easy to accelerate to speeds close to 40 mph. We will also ignore gravity throughout this problem. Otherwise, the Earth (with an escape velocity of [tex]$11 \, \text{km/s}$[/tex]) would have turned into a black hole long ago.

- Part (a)

Suppose that a student wants to go to a restaurant for lunch, but she only has an hour to go, eat, and return in time for class. Considering it usually takes about 30 minutes in most restaurants to get served and eat, what is the farthest restaurant the student can go to without being late for class? Assume the student has a car that can accelerate to its top speed in a negligible amount of time, and the local speed limit is 30 mph.

\[ d_{\max} = 7.500 \text{ miles} \quad \checkmark \text{ Correct!} \]

- Part (b)

The restaurant the student likes to go to doesn't have any clocks. The only way for the student to keep track of time is by using her wristwatch. According to the student's watch, how much time does she actually have for the entire lunch break (travel and eat) if she wants to go to the furthest restaurant and return in time?

\[ t' = 49.84 \text{ min} \quad \checkmark \text{ Correct!} \]

- Part (c)

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

\[ v = \square \, \text{mph} \]

Let's solve Part (c) of the problem step-by-step.

1. Understanding the Situation:

The student wants to bring back ice cream from the restaurant. It is a hot day, and the ice cream will melt in 5 minutes. Therefore, the student needs to return to school within 5 minutes to prevent the ice cream from melting.

2. Return Distance:

Since the farthest restaurant is 7.5 miles away (as found in Part (a)), the student needs to travel this same 7.5 miles back to school.

3. Travel Time Constraint:

The student has a maximum of 5 minutes to cover the return distance to prevent the ice cream from melting.

4. Calculating Speed:

To find the required speed, we need to convert the time from minutes to hours since speed is usually measured in miles per hour (mph). So, 5 minutes is equivalent to [tex]$ \frac{5}{60} $[/tex] hours, which is approximately 0.0833 hours.

5. Formula for Speed:

Speed is calculated using the formula:
[tex]\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\][/tex]

6. Plugging in the Values:

Using the distance (7.5 miles) and the time (0.0833 hours), we can calculate the required speed:
[tex]\[
\text{Speed} = \frac{7.5 \, \text{miles}}{0.0833 \, \text{hours}} \approx 90 \, \text{mph}
\][/tex]

7. Conclusion:

The student will need to drive back at a speed of approximately 90 mph to ensure that the ice cream does not melt before reaching school.