To find the speed of two particles moving towards each other in a relativistic context, we can't simply add their speeds together due to the effects of special relativity. Instead, we use the relativistic velocity addition formula:
\[ v_{rel} = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \( v_{rel} \) is the relative velocity,
- \( u \) and \( v \) are the speeds of the two particles,
- \( c \) is the speed of light (approximately \( 299,792 \) km/s).
In this case, both particles are moving at \( u = 250,000 \) km/s and \( v = 250,000 \) km/s. Substituting these values into the formula, we get:
\[ v_{rel} = \frac{250,000 + 250,000}{1 + \frac{(250,000)(250,000)}{(299,792)^2}} \]
Calculating the denominator:
\[ \frac{(250,000)(250,000)}{(299,792)^2} \approx \frac{62,500,000,000}{89,875,517,873.681} \approx 0.694 \]
So the denominator becomes:
\[ 1 + 0.694 \approx 1.694 \]
Now plugging back into the formula for \( v_{rel} \):
\[ v_{rel} = \frac{500,000}{1.694} \approx 294,000 \text{ km/s} \]
So the two particles will be moving relative to each other at approximately 294,000 km/s.
Thus, the answer is between 250,000 and 500,000 km/s.
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