Question

If astronauts could travel at v disagree. 0.945c, we on Earth would say it takes 4.200 945 4 44 years to reach pha centaur 42Ф Çgint ears away. The astronauts (a) How much time passes on the astronauts' clocks? Your response is within 10% of the correct value. This may be due to roundoff error or you could have a mistake in your calculation. results to at least four-digit accuracy to minimize roundoff error. years arn out all men mediate (b) What is the distance to Alpha Centauri as measured by the astronauts?

Answer 1

Answer:

1.3734 years pass on the astronauts' clock.

The distance measured by astronauts is $1.22*10^{16} \ m$

Explanation:

The fomula for time dilation is:

$t'=t\sqrt{1-\frac{v^2}{c^2} }$

where t' is the time observed by the astronauts, t is the time observed form Earth, v is the velocity of the astronaut and c is the velocity of light in a vacuum.

Replacing our values: t=4.2 year, v=0.945c

$t'=4.2\sqrt{1-\frac{{(0.945c)}^2}{c^2} } =\\\\=4.2\sqrt{1-0.8930 }=4.2*0.3270=1.3734\ years$

The distance measured by astronauts will be their speed multolied by the time is takes them to get to alpha centauri:

$D=v*t'=0.945*c*1.3734=0.945*3.00*10^8 m/s*1.3734*3.154*10^7$

To acomodate fot units we write the speed of light in m/s and the amount of seconds in a year, so the resulting distance will be in meters.$D=1.22*10^{16} \ m$

We should compare this to the distance measured from Earth which is $D=c*t=3.00*10^8 m/s*4.2*3.154*10^7=3.974*10^{16}\ m$