The nearest star to Earth is Proxima Centauri, 4.2 light-years away. 2 parts.. 1

Question

The nearest star to Earth is Proxima Centauri, 4.2 light-years away.
2 parts..
1st ) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 3.7 years, as measured by travelers on the spacecraft?

2nd ) How long does the trip take according to Earth observers?

please show steps with values for each..
like for part 1 I understand the formula to be
L=L0 [1-(v/c)^2]^.5
and Because the star is moving toward the spacecraft, to cover this distance in 4.2 yr, the speed of the
star must be

v=L/t =(4.2/3.7)[1-(v/c)^2]^.5 = 1.135[1-(v/c)^2]^.5

what am i plugging in? or how do i arrange to solve for v..

then I need help with part too as well..

thanks..

Explanation / Answer

L = 4.2 ly, t = 3.7 years a) D = v*t = v*3.7 = 4.2c*sqrt(1-v^2/c^2) v~~0.750359 c to solve: square both sides. v^2*3.7^2=4.2^2*c^2*(1-v^2/c^2) v^2*3.7^2+4.2^2*c^2*v^2/c^2=4.2^2*c^2 v^2*3.7^2+4.2^2*v^2=4.2^2*c^2 so v^2=4.2^2/(4.2^2+3.7^2)*c^2 sqrt both sides v=sqrt(4.2^2/(4.2^2+3.7^2))*c which you can just make the equation btw. v=sqrt(t^2/(t^2+l^2)*c for b. te=T/(1-v^2/c^2)^(1/2)=3.7/(1-(0.75c)^2/c^2)^(1/2)=5.594

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