an enemy spaceship is moving toward your starfighter with a speed of 0.4c, as measured in your reference frame. The enemy ship fires a missile toward you at a speed of 0.7c relative to the enemy ship. If you measure the enemy ship to be 8.0 x 10^6 km away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you ?
5s
To find the time it takes for the missile to reach you, we need to use the concept of time dilation in Special Relativity.
In the given scenario:
- Speed of the enemy spaceship (v_ship) = 0.4c
- Speed of the missile relative to the enemy ship (v_missile) = 0.7c
- Distance between you and the enemy ship (d) = 8.0 x 10^6 km
First, we need to calculate the relative velocity (v_rel) between the missile and you. We can use the relativistic velocity addition formula:
v_rel = (v_missile + v_ship) / (1 + (v_missile * v_ship) / (c^2))
Plugging in the values:
v_rel = (0.7c + 0.4c) / (1 + (0.7c * 0.4c) / (c^2))
v_rel = 1.1c / (1 + 0.28)
v_rel = 1.1c / 1.28
v_rel ≈ 0.8594c
Now, we can calculate the time it takes for the missile to reach you using the formula:
t = d / v_rel
Plugging in the values:
t = (8.0 x 10^6 km) / (0.8594c)
To find the time in your frame, we need to convert the distance from kilometers to a unit of length that matches the speed of light. Since light travels at approximately 3 x 10^5 km/s, we can convert 8.0 x 10^6 km to meters:
d = (8.0 x 10^6 km) * (10^3 m / 1 km)
d = 8.0 x 10^9 m
t = (8.0 x 10^9 m) / (0.8594c)
Now, we can calculate the time:
t = (8.0 x 10^9 m) / (0.8594 * 3 x 10^8 m/s)
t ≈ 30 seconds
Therefore, it will take approximately 30 seconds, as measured in your frame, for the missile to reach you.