Basic Special Relativity (Time Dilation)

[bmb2009]

Homework Statement

The Concorde traveled 8000km between two places with an average speed of 375 m/s. What is the time difference between two atomic clocks, one on the train and one at rest with respect to the train?

Homework Equations

T=AT' where A is the Lorentz gamma factor

The Attempt at a Solution

Seemed simple enough the distance divided by time would yield the T (the time taken from the rest frame) so (8x10^6 m/s)/(375) and then solve for T' (proper time which would be the time of a clock on the train correct?) but the speed of 375 m/s is so miniscule in comparison to the speed of light the lorentz factor comes out to be 1 (which isn't surprising) but in the back of the book it says the answer is 16.7 microseconds...what am I doing wrong? Thanks

 

[TSny]

To a good approximation for your problem, you can approximate [1-(v/c)²]^1/2 using the binomial approximation (1-x)^a≈1-ax for x<<1.

 

[cepheid]

Homework Statement

The Concorde traveled 8000km between two places with an average speed of 375 m/s. What is the time difference between two atomic clocks, one on the train and one at rest with respect to the train?

Homework Equations

T=AT' where A is the Lorentz gamma factor

The Attempt at a Solution

Seemed simple enough the distance divided by time would yield the T (the time taken from the rest frame) so (8x10^6 m/s)/(375) and then solve for T' (proper time which would be the time of a clock on the train correct?) but the speed of 375 m/s is so miniscule in comparison to the speed of light the lorentz factor comes out to be 1 (which isn't surprising) but in the back of the book it says the answer is 16.7 microseconds...what am I doing wrong? Thanks

Not using a calculator with enough precision?

 

[bmb2009]

To a good approximation for your problem, you can approximate [1-(v/c)²]^1/2 using the binomial approximation (1-x)^a≈1-ax for x<<1.

How does this help though? that just makes 1-ax = 1 - (7.8125e-13) which still is very close to 1. Where/how do you produce the 16.7 nanoseconds?

 

[TSny]

Use the binomial approximation in the equation T = AT' without plugging any numbers in yet. Then see if you can rearrange for the quantity T-T'. Then plug in numbers.

 

[bmb2009]

Use the binomial approximation in the equation T = AT' without plugging any numbers in yet. Then see if you can rearrange for the quantity T-T'. Then plug in numbers.

I don't think you can )easily) solve for T-T' with the expansion... T=T'/1-ax ==> T-Tax=T'...unless I am missing some simply algebra i don't see how to manipulate to solve for T-T'

 

[TSny]

I don't think you can )easily) solve for T-T' with the expansion... T=T'/1-ax ==> T-Tax=T'...unless I am missing some simply algebra i don't see how to manipulate to solve for T-T'

Rearrange your last equation as T - T' = Tax

As you said in your original post, you know how to get T for the right hand side.

 

[bmb2009]

Rearrange your last equation as T - T' = Tax

As you said in your original post, you know how to get T for the right hand side.

ahhh finally got it.. Thanks a bunch!