The velocity of a plane is 0.80c (c is speed of light), a missile if fired from the plane parallel to its direction of motion at 0.60c. What is the velocity of the missile as seen from the Earth.
At low velocities it is possible to add velocities u and v measured by people in two different inertial frames of reference as R = u + v. For example if a person standing on the ground measures the velocity of a car as u and a person sitting in the car measures the velocity of a bee in the car as u, the velocity of the bee as measured by the person on the ground can be approximated very closely as R = u + v. But as the velocities approach the speed of light, it is not possible to do that.
The Special Theory of Relativity shows that the distance and time measured in different reference frame is not the same. Due to this it is not possible to simply add velocities of objects measured across different reference frames. The formula to find the composite velocity of u and v becomes R = (u + v)/(1 + u*v/c^2) where c is the velocity of light.
Here, the velocity of the plane is 0.8c and the velocity of the missile fired as measured by a person on the plane is 0.6c. The velocity of the missile as measured by a person on the Earth would not be 1.4c but instead it would be R = (0.8c+0.6c)/(1+0.8*0.6c^2/c^2) = 1.4c/(1.48) = 0.95c
The velocity of the missile as measured from the Earth is 0.95c.
The equation for relativistic velocity is given by
W = (U + V)/(1 + UV/c^2)
W is the apparent speed as seen from the proper reference point on the surface of the Earth.
U is the speed of the plane
V is the speed of the missile relative to the plane
c is the speed of light
W = (0.80c + 0.60c)/(1 + 0.80c*0.60c/c^2)
= 1.40c/(1 + 0.48c^2/c^2)
W = 0.95c