How fast must the spacecraft travel? Special Relativity

[VaccumEnergy]

Homework Statement

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1. Homework Statement
A spacecraft is prepared to visit α-Centauri, which is at the distance 4.37 light years from the Sun. Provisions for the crew are prepared for the duration of 16 years. How fast must the spacecraft travel for this provision to be enough?
Answer this question in two ways by considering:
(i) time dilation
(ii) length contraction. (The period of acceleration, dece- laration, turnaround and visit are neglected compared to the whole time of the travel)

Homework Equations

Time dilation:
t = gamma*t0, where t is the time from the Earths inertial frame and t0 is the 'proper time' i.e. time perceived from Earth on the rocket.

Length Contraction:
l=l0/gamma, where l is the length perceived from an inertial frame of a body moving with speed v relative to it of length l, l0 is length perceived when v = 0

gamma = 1/(1-v^2/c^2)^-1/2, v is relative velocity and c is speed of light.

The Attempt at a Solution

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I'm unsure how to begin this question, been stuck for almost an hour. Making me feel stupid, please helppppp

 

[jbriggs444]

Let's work on the "time dilation" approach.

You have an equation $t = {gamma}*t_0$

You know from the problem statement that t (the time elapsed according to the spacecraft clocks must be no more than 16 years.

Can you calculate $t_0$ (the time taken for the trip according to Earth clocks) in terms of the distance and the spacecraft velocity?

Can you expand $t = {gamma}*t_0$ in terms of the spacecraft velocity and the distance from Earth to Alpha Centauri?

 

[VaccumEnergy]

Let's work on the "time dilation" approach.

You have an equation $t = {gamma}*t_0$

You know from the problem statement that t (the time elapsed according to the spacecraft clocks must be no more than 16 years.

Can you calculate $t_0$ (the time taken for the trip according to Earth clocks) in terms of the distance and the spacecraft velocity?

Can you expand $t = {gamma}*t_0$ in terms of the spacecraft velocity and the distance from Earth to Alpha Centauri?

The proper time, on the spacecraft , is given by:

$t_0 = t / {gamma}$

The proper time must be less than $16[years]$, is a constraint (i.e. $t/{gamma} < 16 {years}$)
It then follows that :

$t/{gamma} = (2*4.37*c*years)/(v*{gamma}) < 16 years$

Now solving for $v/c$ gives :

$v/c >0.48 (approx)$Does that look okay?

 

[jbriggs444]

It is always wise to do a sanity check on results like this. Scribbling on the back of an envelope...

Your computed result is a v/c of 0.48. That means that the trip will take 4.38 light years / 0.48 = 9.125 Earth years one way. 18.25 years round trip. This is dilated by a gamma of 1.14 and the result is 16 years of elapsed shipboard time, as desired.

Yes, that looks right.