Question
What is the relative velocity of two particles moving towards each other at 200,000 km/s each, using the relativistic velocity addition formula?
Answers
Answered by
GPT-4o mini
To find the relative velocity of two particles moving towards each other at high speeds using the relativistic velocity addition formula, we can use the following formula:
\[
u' = \frac{u + v}{1 + \frac{uv}{c^2}}
\]
where:
- \( u' \) is the relative velocity of one particle as observed from the other.
- \( u \) is the velocity of one particle.
- \( v \) is the velocity of the other particle (in the opposite direction).
- \( c \) is the speed of light, approximately \( 299,792 \) km/s (or \( 300,000 \) km/s for rough calculations).
Given:
- \( u = 200,000 \) km/s (velocity of particle A towards particle B)
- \( v = -200,000 \) km/s (velocity of particle B towards particle A)
We can substitute these values into the formula:
\[
u' = \frac{200,000 + (-200,000)}{1 + \frac{(200,000)(-200,000)}{c^2}}
\]
This simplifies to:
\[
u' = \frac{200,000 - 200,000}{1 - \frac{200,000^2}{c^2}} = \frac{0}{1 - \frac{200,000^2}{c^2}}
\]
Next, we need to compute \( \frac{200,000^2}{c^2} \):
To get approximations:
- \( c^2 \approx (300,000 \text{ km/s})^2 = 90,000,000,000 \text{ km}^2/\text{s}^2 \)
- \( 200,000^2 = 40,000,000,000 \text{ km}^2/\text{s}^2 \)
Thus:
\[
\frac{200,000^2}{c^2} = \frac{40,000,000,000}{90,000,000,000} \approx 0.4444
\]
Now substituting back into the relative velocity formula:
\[
u' = \frac{0}{1 - 0.4444} = \frac{0}{0.5556} = 0
\]
Now this leads to a contradiction because we should expect a non-zero value. Thus we need to actually compute assuming they are moving towards each other:
Using \( u = 200,000 \) km/s and \( v = 200,000 \) km/s with both being the same direction but negating each other in relative motion to make direction terms meaningful, we get:
\[
u' = \frac{200,000 + 200,000}{1 + \frac{200,000 \cdot 200,000}{c^2}}
\]
Where \( c = 300,000 \text{ km/s} \). So let's substitute that value back in:
\[
= \frac{400,000}{1 + \frac{40,000,000,000}{90,000,000,000}}
\]
Calculating the denominator:
\[
= 1 + 0.4444 = 1.4444
\]
Finally,
\[
u' = \frac{400,000}{1.4444} \approx 276,000 \text{ km/s}
\]
Thus, the relative velocity of the two particles moving towards each other, when properly taking relativistic effects into account, is approximately **276,000 km/s**.
\[
u' = \frac{u + v}{1 + \frac{uv}{c^2}}
\]
where:
- \( u' \) is the relative velocity of one particle as observed from the other.
- \( u \) is the velocity of one particle.
- \( v \) is the velocity of the other particle (in the opposite direction).
- \( c \) is the speed of light, approximately \( 299,792 \) km/s (or \( 300,000 \) km/s for rough calculations).
Given:
- \( u = 200,000 \) km/s (velocity of particle A towards particle B)
- \( v = -200,000 \) km/s (velocity of particle B towards particle A)
We can substitute these values into the formula:
\[
u' = \frac{200,000 + (-200,000)}{1 + \frac{(200,000)(-200,000)}{c^2}}
\]
This simplifies to:
\[
u' = \frac{200,000 - 200,000}{1 - \frac{200,000^2}{c^2}} = \frac{0}{1 - \frac{200,000^2}{c^2}}
\]
Next, we need to compute \( \frac{200,000^2}{c^2} \):
To get approximations:
- \( c^2 \approx (300,000 \text{ km/s})^2 = 90,000,000,000 \text{ km}^2/\text{s}^2 \)
- \( 200,000^2 = 40,000,000,000 \text{ km}^2/\text{s}^2 \)
Thus:
\[
\frac{200,000^2}{c^2} = \frac{40,000,000,000}{90,000,000,000} \approx 0.4444
\]
Now substituting back into the relative velocity formula:
\[
u' = \frac{0}{1 - 0.4444} = \frac{0}{0.5556} = 0
\]
Now this leads to a contradiction because we should expect a non-zero value. Thus we need to actually compute assuming they are moving towards each other:
Using \( u = 200,000 \) km/s and \( v = 200,000 \) km/s with both being the same direction but negating each other in relative motion to make direction terms meaningful, we get:
\[
u' = \frac{200,000 + 200,000}{1 + \frac{200,000 \cdot 200,000}{c^2}}
\]
Where \( c = 300,000 \text{ km/s} \). So let's substitute that value back in:
\[
= \frac{400,000}{1 + \frac{40,000,000,000}{90,000,000,000}}
\]
Calculating the denominator:
\[
= 1 + 0.4444 = 1.4444
\]
Finally,
\[
u' = \frac{400,000}{1.4444} \approx 276,000 \text{ km/s}
\]
Thus, the relative velocity of the two particles moving towards each other, when properly taking relativistic effects into account, is approximately **276,000 km/s**.