Question on kinematics and theory of relativity

Question

If two particles A and B travel in opposite directions with the velocity of 0.6c and 0.7c respectively, the particle A sees particle B moving with a speed of 1.3c in the opposite direction (c is the speed of light). Is this possible?

Answer 1

What you ask is in fact one of the problems that gave birth to our understanding of special relativity. In Newtonian Mechanics, a.k.a. everyday physics you would be correct. If an objetct is traveling with $0.7v$ and it encounters another one that travels in opposite direction with $0.6v$, then the total velocity would simply be the sum of the tho, which is $1.3v". This is what we call a "Galilean transformation"

This agrees with all of our observations in our daily lives, and we take it for granted. However, in the 20th century scientists began to notice that many aspects of our understanding of physics where flawed when velocities start getting very high. That means, when they can be compared to the speed of light.

You see, one of the laws of physics states that the speed of light cannot be surpassed. However, as you show, Galilean transformations show that if we had opposing velocities of $0.6v$ and $0.7v$ we should obtain $1.3v$, which simply cannot be. The problem was solved by Henrik Lorentz, who proposed what we now call the "Lorentz transformations". According to them, the velocities are not added linearly, but as follows: $$u'=\frac{u+v}{1+\frac{u*v}{c^2}}$$ Given $u=0.6c$ and $v=0.7v$, we obtain a velocity of

$$u'=\frac{1.3c}{1+\frac{0.6c*0.7c}{c^2}}=\frac{1.3c}{1.42}=0.915c$$

This is the correct way to add velocities. You can notice that in the case were both $u$ and $v$ are much smaller than the speed of light $c$, the second term in the denominator can be neglected, and you end up with the classical case.

Answer 2

This is not the case, each will measure the opposite spacecraft to have a speed $v$ satisfying $v<c$. This phenomenon is described by Einstein's theory of special relativity. Consider reading Spacetime Physics An Introduction to Relativity by Taylor and Wheeler for a good introduction to this concept.