Missiles are fired towards the earth from a spacecraft. The missiles are moving with a speed of 0.8c with respect to the spacecraft. If the spacecraft itself has a speed of 0.3c with respect to Earth, how fast are the missiles observed to travel with respect to Earth? What would be the speed of the missiles as predicted by Galilean transformation? Comment on this result.
To determine the speed of the missiles observed from Earth, we need to use the relativistic velocity addition formula. According to special relativity, the formula for adding velocities is given by:
v = (v1 + v2)/(1 + (v1*v2)/c^2)
where v is the final velocity, v1 is the velocity of the missiles with respect to the spacecraft (0.8c), v2 is the velocity of the spacecraft with respect to Earth (0.3c), and c is the speed of light.
Plugging in the values into the formula:
v = (0.8c + 0.3c)/(1 + (0.8c)*(0.3c)/(c^2))
= (1.1c)/(1 + 0.24)
= (1.1c)/(1.24)
= 0.8871c
Therefore, the speed of the missiles observed from Earth would be approximately 0.8871 times the speed of light (c).
Now, let's consider the Galilean transformation to calculate the velocity. According to the Galilean transformation, velocities simply add up:
v = v1 + v2
Using the same values as before:
v = 0.8c + 0.3c
= 1.1c
According to the Galilean transformation, the speed of the missiles as predicted would be 1.1 times the speed of light (c).
Comparing the two results, we can see that with the relativistic velocity addition formula, the speed of the missiles is less than the speed predicted by the Galilean transformation. This is because special relativity takes into account the relativistic effects at high speeds, where the Galilean transformation fails to provide accurate predictions. The relativistic addition formula correctly accounts for time dilation and length contraction at high speeds and hence provides a more accurate result.