An object whose mass at rest is 3g is traveling at 4/5 the speed of light when it collides head-on with an identical object going in the opposite direction at the same speed. If the two masses stick together and no energy is radiated away, what is the mass of the composite object?
The total energy before the collision should be the same after the collision because no
energy is radiated away. For the total energy the equation is
\( E = \gamma mc^2 \)
The value of gamma before the collision is the same for both particles:
\( \gamma = \dfrac{1}{\sqrt{1-v^2/c^2}} = \dfrac{1}{\sqrt{1-0.8^2}} = \dfrac 53 \)
After the collision the value of \(\gamma\) is 1, because the velocity is zero. Then
\( \dfrac53 (2 \times 3\mathrm g) c^2= Mc^2 \rightarrow M=10 \mathrm g\)
I should mention that in principle this problem is a good practice of the equations and the calculations
that you need to learn in relativity, but I do not think this is something that you could do in a lab
anytime soon. The problem is completely made up and perhaps not possible at all.
Einstein used to like things like this and called them "Gedanken Experiments".