Question

Two particles ina particle accelerator are moving tdowards each other. They each have a speed of 250,000km/s relative to the ground. How fast will they be moving relative to each other?
500,000 km/s
between 250,000 and 500,000 km/s

Answer 1

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**.