Time Dilation Calculation: Astronaut vs. Earth Observer | Homework Solution

[wilson_chem90]

Homework Statement

An astronaut traveling at 0.90 c, with respect to Earth, measures his pulse and finds it to be 70 beats per minute.

a) Calculate the time required for one pulse to occur, as measured by the astronaut.

b) Calculate the time required for one pulse to occur, as measured by an Earth-based observer.

c) Calculate the astronauts pulse, as measured be an Earth-based observer.

Homework Equations

t = to/{1 - v^2/c^2} ({ } = square root)

The Attempt at a Solution

a) 70 beats per minute/60 seconds = 1.167 beats/per second
So that would mean it would take 1.167 seconds for one beat to occur.

b) t = to/{1 - v^2/c^2}
= 1.167 seconds {1 - (.90 c)^2/c^2}
= 1.167 seconds/(0.43589)
= 2.68 seconds
So it would take 2.68 seconds for once pulse to occur, measured by an Earth observer.

c) 2.68 seconds/beat x 60 seconds
= 160.8 beats/minute

It would be approx. 161 beats/minute if it was measured by an Earth observer.

I'm not sure if this is correct, can someone please confirm? Thank you

 

[kuruman]

(a) Is not correct. If there are 70 beats in a minute and 60 seconds in a minute, then there are more beats than seconds in a minute. Therefore the time between beats is less than one second. Try again and reason it out correctly.

 

[wilson_chem90]

a) 60 seconds/ 70 beats per minute = 0.857 beats/second.

b)t = to/{1 - v^2/c^2}
= 0.857 seconds {1 - (.90 c)^2/c^2}
= 0.857 seconds/(0.43589)
= 0.373 seconds
So it would take 0.373 seconds for once pulse to occur, measured by an Earth observer.

c) 0.373 seconds x 60 seconds = 22.4 beats/minute

is that right?

 

[kuruman]

As measured by the Earth observer, is the time between pulses longer or shorter than 0.857 s? Gamma is greater than one, so if "longer", you multiply by gamma; if "shorter" you divide by gamma.

 

[wilson_chem90]

Well it should be shorter

 

[wilson_chem90]

sorry it would be longer, not shorter

 

[kuruman]

If it's shorter, then according to the Earth observer the astronaut's heart beats faster which means that the astronaut ages faster than the Earth observer. Does that sound right?

 

[wilson_chem90]

ohh i multiplied by accident. I was supposed to divide it. it would be 1.97 seconds/beat. Man i am just not cluing in today, sorry.

 

[wilson_chem90]

then c) would be 30.4 beats/minute. And no it wouldn't make sense at all if it was shorter.

 

[wilson_chem90]

is that correct?

 

[kuruman]

I didn't do the numbers, but it looks about right.

 

[wilson_chem90]

oh okay, thanks for your help, i really appreciate it