- #1
C_Ovidiu
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A person on Earth observes two rockets moving directly toward each other and colliding. At time t=0 in the Earth frame observer determines that rocket A, traveling to the right at [tex]v_a=0.8c[/tex] is at point a and rocket B traveling to the left at [tex] v_b=0.6c[/tex] is at point b. They are separated by a distance [tex]l=4.2*10^8[/tex].
How much time will elapse in A's frame of reference until the two rockets collide?
1. If we consider the time that will elapse in Earth frame until collision and then transform this interval in A's frame because of time dilation then we obtain :
[tex]t_A=\frac{l\cdot\sqrt{1-\beta_a^2}}{v_a+v_b}=0.6s[/tex]
2. Howerer, if we change our approach in the following way.
we transform the distance l between the two ships at [tex]t=t_A=0 [/tex] :
[tex]l_a=\frac{l}{\sqrt{1-\beta_a^2}}[/tex]
we consider then the speed of B in A's frame of reference [tex]v_b^a=-\frac{v_b+v_a}{1+\frac{v_bv_a}{c^2}}[/tex]
Now we get [tex]t_A=\frac{l_a}{v_b^a}[/tex] . As you can see this rationing yields a totally different result. What is wrong in this second approach?
How much time will elapse in A's frame of reference until the two rockets collide?
1. If we consider the time that will elapse in Earth frame until collision and then transform this interval in A's frame because of time dilation then we obtain :
[tex]t_A=\frac{l\cdot\sqrt{1-\beta_a^2}}{v_a+v_b}=0.6s[/tex]
2. Howerer, if we change our approach in the following way.
we transform the distance l between the two ships at [tex]t=t_A=0 [/tex] :
[tex]l_a=\frac{l}{\sqrt{1-\beta_a^2}}[/tex]
we consider then the speed of B in A's frame of reference [tex]v_b^a=-\frac{v_b+v_a}{1+\frac{v_bv_a}{c^2}}[/tex]
Now we get [tex]t_A=\frac{l_a}{v_b^a}[/tex] . As you can see this rationing yields a totally different result. What is wrong in this second approach?