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

1.3734 years pass on the astronauts' clock.

The distance measured by astronauts is
`1.22*10^(16) \ m`

Step-by-step explanation:

The fomula for time dilation is:

`t'=t\sqrt{1-(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√(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`