A large asteroid is heading directly towards Earth. If a missile travels at a speed of 0.800c, what should the timer be set at so that the explosive detonates just as the missile impacts the asteroid?

To determine the appropriate timer setting for the missile to detonate upon impact with the asteroid, we need to consider the effects of relativistic speeds, specifically how time dilation can affect the perceived time for the missile traveling at a speed of 0.800c, where c is the speed of light.

At 0.800c, the time experienced by the missile will not be the same as that experienced by an observer on Earth due to time dilation, described by the Lorentz factor (γ), which is calculated as:

γ = 1 / √(1 - v²/c²)

Substituting in the speed of the missile:

v = 0.800c → v² = (0.800)²c² = 0.64c²

Now, calculating γ:

γ = 1 / √(1 - 0.64) = 1 / √(0.36) = 1 / 0.6 = 5/3 or approximately 1.667.

This factor indicates how much time is dilated for the missile's journey compared to an observer on Earth.

If the missile is set to detonate just as it impacts the asteroid, we need to calculate the time that elapses