Calculating Speed and Time Between Two Vehicles Heading to Mars

In summary, the conversation discusses the concept of time dilation in the context of a vehicle travelling towards Mars at a constant speed. The conversation also touches on the proper time and length in different frames of reference. The first question asks for the time on the timer of a superspeeder when it has travelled a certain distance past the vehicle, and the second question asks for the time on the vehicle's timer when the superspeeder reads the values calculated in the previous question. The formula t = t0/(sqrt(1-(v/c)^2)) is used to find the proper time, and the formula t = d/v is used to calculate the time according to the vehicle's frame of reference.
  • #1
jono90one
28
0

Homework Statement



You are in a vehicle heading towards Mars at constant speed. A superspeeder is going at constant speed 0.9c relative to you. When the superspeeder pases you, both times start at 0.

"At the point when you measure the superspeeder has traveled 1.4 x 10^7 m past you, what does teh superspeeder read on their timer"

"At the point when the superspeeder reads the values calculated in previous question, what do you read on yours?"


Homework Equations


t=t0 gamma
s=d/t

The Attempt at a Solution


The first one i get 0.5 by just using speed = distance / time and re-aranging

But, I am sure which is t and t0, t0 is the proper time, but t0 is always smaller than t, so teh thing going faster must always be t0 right?

So is t0 here the time the superspeeder reads?
(As i was told it was the other way around which has confused me :S)


Also for clarity, is t0 and l0 measured in the same frame of reference?

Thanks.
 
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  • #2
jono90one said:
The first one i get 0.5 by just using speed = distance / time and re-aranging
That will tell you the time according to you, not according to the superspeeder. (But redo your calculation.)

But, I am sure which is t and t0, t0 is the proper time, but t0 is always smaller than t, so teh thing going faster must always be t0 right?
Assuming you're talking about the time dilation formula, t0 is the proper time and is always the smaller number.

So is t0 here the time the superspeeder reads?
Yes.

Also for clarity, is t0 and l0 measured in the same frame of reference?
What is l0 ?
 
  • #3
lo is the proper length - not to do with this question but wanted to clarify l0 is in the same frame of reference has t0.

Well from what i understand there, i have started one correctly, but need to extend it using t = t0/(sqrt(1-(v/c)^2))

To find t0


Then for 2) it would simply be t? (which is t=d/v from 1), seems a little silly to word the questions this way if so)

also for t = d/v i got 0.05 (2dp) my bad.

Or have i miss understood?
 
  • #4
jono90one said:
lo is the proper length - not to do with this question but wanted to clarify l0 is in the same frame of reference has t0.
Yes. Example: A rocketship moving along. The proper length of the ship is its length as measured in its own frame. Similarly, the proper time is the time as measured by a clock in its own frame.

Well from what i understand there, i have started one correctly, but need to extend it using t = t0/(sqrt(1-(v/c)^2))

To find t0
Good.


Then for 2) it would simply be t? (which is t=d/v from 1), seems a little silly to word the questions this way if so)
Yes. (The parts would make more sense if given in the reverse order.)

also for t = d/v i got 0.05 (2dp) my bad.
Good.
 
  • #5


I would approach this problem by first clarifying the variables and equations being used. It seems that the equation being used here is the time dilation equation, t=t0/gamma, where t is the time measured by the observer and t0 is the proper time measured by the object in motion. Gamma is the Lorentz factor, which is dependent on the relative speeds of the two objects.

In this case, the superspeeder is traveling at 0.9c relative to the observer. So, we can calculate the Lorentz factor as gamma = 1/sqrt(1-(v^2/c^2)) = 1/sqrt(1-(0.9c^2/c^2)) = 1/sqrt(0.19) = 1/0.436 = 2.29.

Using the time dilation equation, we can now calculate t0 for the superspeeder when it has traveled 1.4 x 10^7 m past the observer. t0 = t/gamma = 0.5/2.29 = 0.218 seconds. So, at this point, the superspeeder would read 0.218 seconds on its timer.

For the second question, we need to use the time dilation equation again, but this time to calculate t, the time measured by the observer. We know t0 = 0.218 seconds and gamma = 2.29. So, t = t0 * gamma = 0.218 * 2.29 = 0.5 seconds. Therefore, at the point when the superspeeder reads 0.218 seconds, the observer would read 0.5 seconds on their timer.

To answer the last question, t0 and t are measured in different frames of reference. t0 is measured in the frame of reference of the superspeeder, while t is measured in the frame of reference of the observer. This is why we need to use the time dilation equation to convert between the two frames.

In summary, the time dilation equation allows us to calculate the time measured by an object in motion relative to an observer in a different frame of reference. It is important to clearly define the variables being used and understand the concept of frames of reference when solving problems like these.
 

1. How do you calculate the speed of two vehicles heading to Mars?

To calculate the speed, you would need to know the distance between the two vehicles and the time it takes for them to reach Mars. The formula for speed is distance divided by time.

2. What units are typically used to measure speed when calculating the time between two vehicles heading to Mars?

The most commonly used units to measure speed are kilometers per hour (km/h) or miles per hour (mph). However, for space travel, scientists may also use units such as kilometers per second (km/s) or miles per second (mph).

3. How does the distance between two vehicles affect the time it takes for them to reach Mars?

The greater the distance between the two vehicles, the longer it will take for them to reach Mars. This is because they will need to travel a longer distance, which will require more time.

4. Can you calculate the time between two vehicles heading to Mars without knowing the speed?

No, knowing the speed is necessary to calculate the time between two vehicles heading to Mars. Without the speed, you would only have one variable in the formula and cannot solve for time.

5. What are some factors that may affect the accuracy of calculating the time between two vehicles heading to Mars?

Factors that may affect the accuracy of the calculation include changes in speed during the journey, gravitational forces from other celestial bodies, and potential technical malfunctions or errors in data collection. It is important to continuously monitor and adjust for these factors to ensure the most accurate calculation.

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