A rocket fires a projectile at a speed of 0.95c (relative to the rocket) while traveling past the Earth. An earthbound scientist measures the projectile's speed to be 0.90c. What was the rocket's speed as a fraction of c?
so what is the contraction factor?
Well, well, well, we've got ourselves a speedy rocket! Let's dig into this spacey situation, shall we?
Now, according to Einstein's theory of special relativity, the speed of light (c) is a constant for all observers. So, we have a projectile being fired at a speed of 0.95c "relative to the rocket," and an earthbound scientist measures it to be moving at 0.90c.
To figure out the rocket's speed, we need to do a little mathematical dance. So, let's assume that the rocket's speed (v) is what we're after.
Now, the formula for combining velocities in special relativity is not as straightforward as 1+1=2. Instead, it's a bit funky. It goes like this:
v = (v1 + v2) / (1 + v1*v2/c^2)
Using this formula, with v1 = 0.95c (the projectile's speed relative to the rocket) and v2 = 0.90c (the projectile's speed as measured by the earthbound scientist), we can calculate the rocket's speed (v).
So, let's plug those numbers in, my friend!
v = (0.95c + 0.90c) / (1 + (0.95c * 0.90c)/c^2)
v = (1.85c) / (1 + 0.855)
v = (1.85c) / (1.855)
v ≈ 0.99594c
So, the rocket's speed, as a fraction of c, is approximately 0.99594c. Quite a zippy rocket, if you ask me!
I hope that answer launched a few giggles! Feel free to ask me anything else - I'm here to clown around!
To solve this problem, we can use the relativistic velocity addition formula. The formula for adding velocities in special relativity is:
V = (v + u) / (1 + (vu / c^2))
Where:
V = measured velocity (0.90c)
v = projectile's velocity relative to Earth (unknown)
u = rocket's velocity relative to Earth (unknown)
c = speed of light in vacuum (constant = 299,792,458 m/s)
Using this formula, we can substitute the known values and solve for v:
0.90c = (0.95c + u) / (1 + (0.95c * u / c^2))
Simplifying the formula:
0.90c = (0.95c + u) / (1 + 0.95u / c)
Cross-multiplying:
(0.90c) * (1 + 0.95u / c) = 0.95c + u
0.90c + 0.855u = 0.95c + u
Rearranging the equation:
u - 0.855u = 0.95c - 0.90c
0.145u = 0.05c
Dividing both sides by 0.145:
u = (0.05c) / 0.145
Simplifying:
u = 0.3448c
Therefore, the rocket's speed was approximately 0.3448 times the speed of light (c).
To solve this problem, we need to make use of the relativistic addition of velocities, which is given by the formula:
v' = (v + u) / (1 + (vu / c^2))
Where:
- v' is the measured velocity of the projectile as observed by the earthbound scientist,
- v is the velocity of the projectile relative to the rocket,
- u is the velocity of the rocket relative to the Earth,
- c is the speed of light in a vacuum.
In this case, we know that:
v' = 0.90c (the measured velocity of the projectile as observed by the earthbound scientist),
v = 0.95c (the velocity of the projectile relative to the rocket).
We want to find u, the velocity of the rocket relative to the Earth.
By substituting the given values into the equation, we have:
0.90c = (0.95c + u) / (1 + (0.95c * u / c^2))
Simplifying further, we have:
0.90c = (0.95c + u) / (1 + 0.95u / c)
Next, let's multiply both sides of the equation by (1 + 0.95u / c) to eliminate the denominator:
0.90c * (1 + 0.95u / c) = 0.95c + u
Expanding the left side of the equation, we get:
0.90c + 0.855u = 0.95c + u
Now, let's isolate the variable u on one side of the equation:
0.855u - u = 0.95c - 0.90c
-0.145u = 0.05c
Finally, solving for u:
u = (0.05c) / (-0.145)
Dividing both the numerator and denominator by -0.145:
u = - (0.05c) / 0.145
Therefore, the speed of the rocket, u, as a fraction of c, is approximately -0.345c (negative because it's moving in the opposite direction of the projectile).