Question

Two particles in a particle accelerator are moving towards each other. They each have a speed of 250,000 km/s relative to the ground. How fast will they be moving relative to each other? (1 point)
Responses

Between 250,000 and 500,000 km/s
Between 250,000 and 500,000 km/s

250,000 km/s
250,000 km/s

Less than 250,000 km/s
Less than 250,000 km/s

500

Answer 1

To find the speed of one particle relative to the other when both are moving towards each other at high speeds, we need to use the relativistic velocity addition formula. According to this formula, if two objects move towards each other with velocities \( u \) and \( v \), the relative velocity \( w \) is given by:

\[
w = \frac{u + v}{1 + \frac{uv}{c^2}}
\]

where \( c \) is the speed of light (approximately \( 300,000 \) km/s).

In this case, both particles are moving at \( u = 250,000 \) km/s and \( v = 250,000 \) km/s. Plugging in the values:

\[
w = \frac{250,000 + 250,000}{1 + \frac{(250,000)(250,000)}{(300,000)^2}}
\]
\[
w = \frac{500,000}{1 + \frac{62,500,000,000}{90,000,000,000}}
\]
\[
= \frac{500,000}{1 + 0.694444}
\]
\[
= \frac{500,000}{1.694444}
\]
\[
\approx 295,000 \text{ km/s}
\]

Thus, the relative speed of the two particles is less than 500,000 km/s, and it is also less than 250,000 km/s. Therefore, the correct answer is:

**Less than 250,000 km/s.**