Special Relativity Problem?
1) How fast would you have to move relative to a meter stick for its length to measure 97.2cm in your frame of reference?
2) At what speed will the Newtonian expression for momentum be in error by 5%?
3) Earth and Sun are 8.2 light minutes apart, as measured in their rest frame. What is the speed of a spacecraft that make the trip in 4.7min according to its on-board clocks?
Comments
1) L = L₀/γ
L=0.972 m
L₀ = 1 m
γ=1/sqrt(1-(v/c)^2)
Solve for v (I get roughly .234c).
2) γmv=mv1.05
mv cancels out, so now the equation becomes γ=1.05, and you can solve for v (I get roughly .305c).
3) L₀=60*8.2*3*10^8 = 1.476*10^11 m
t₀ = 4.7*60 = 282 sec
v=L₀/t = L/t₀
γ=t/t₀
Plug L₀/t into the gamma equation (here γ=t/t₀) for v and solve for t. I'll help you out:
1/sqrt(1-(L₀/(tc))^2)=t/t₀
1-(L₀/(tc))^2 = (t₀/t)^2
t^2 = (L₀/c)^2 + (t₀)^2
t = sqrt((L₀/c)^2 + (t₀)^2) = 576.1 sec
Now plug this value for t into the velocity equation above:
v = L₀/t =0.87c or so