Answer to Question #89212 in Mechanics | Relativity for Sridhar

Question

Two particle comes towards each other with a speed 0.8c with respect to the laboratory their relative speed is

Answer 1

According to the special theory of relativity and the composition law for velocities, there exists the following relation (below we consider the case of a 1-D motion along the x-axis for simplicity):

$$u'_x = \frac{u_x - v}{1- \frac{u_x v}{c^2}},$$

where u_x and u_x' are the velocities of an object relative to the laboratory and moving frames correspondingly, v is the velocity of a moving frame relative to the laboratory frame. Let us consider the case when the moving frame is related to the 2nd particle (flying towards the x-axis negative direction). In this case the relative velocity of the 1st particle in respect to the 2nd one is:

$$u_x = 0.8c, \quad v = -0.8c$$

$$u'_x = \frac{0.8c + 0.8c}{1 + \frac{0.8c \cdot 0.8c }{c^2}} = \frac{1.6}{1.64}c \approx 0.98 c$$

Answer: 0.98c.