A spaceship approaching a planet at a speed of 0.20c fires a probe toward a planet. Observers on the planet see the probe approaching at a speed of 0.61c. What is the speed of the probe as measured from the spaceship
To calculate the speed of the probe as measured from the spaceship, we can use the relativistic velocity addition formula.
The formula for adding velocities in special relativity is:
v = (v1 + v2)/(1 + v1*v2/c^2)
Where:
v = velocity of the probe as measured from the spaceship
v1 = velocity of the probe as measured from the planet (0.61c)
v2 = velocity of the spaceship as measured from the planet (0.20c)
c = speed of light in a vacuum (approximately 3.00 x 10^8 meters per second)
Plugging in the values into the formula:
v = (0.61c + 0.20c)/(1 + (0.61c * 0.20c)/(3.00 x 10^8)^2)
= (0.81c)/(1 + (0.12c^2)/(9.00 x 10^16))
= (0.81c)/(1 + 0.12c^2/(9.00 x 10^16))
Now we can simplify the expression:
v = (0.81c)/(1 + 0.12c^2/(9.00 x 10^16))
= (0.81c)/(1 + 0.12/(9.00 x 10^16/c^2))
= (0.81c)/(1 + 0.12/(9.00 x 10^16/9.00 x 10^16)) [since c^2 = 9.00 x 10^16]
= (0.81c)/(1 + 0.12)
= (0.81c)/(1.12)
Calculating the value:
v ≈ 0.7232c
Therefore, the speed of the probe as measured from the spaceship is approximately 0.7232 times the speed of light (c).