To determine the speed of two particles moving towards each other, we need to use the relativistic velocity addition formula since they are moving at speeds close to the speed of light. The formula for the relativistic addition of velocities \( u \) and \( v \) is given by:
\[ w = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \( w \) is the relative speed of one particle as observed from the other,
- \( u \) and \( v \) are the speeds of the individual particles,
- \( c \) is the speed of light (approximately \( 300,000 \) km/s).
In this case:
- \( u = 250,000 \) km/s,
- \( v = 250,000 \) km/s.
Let's calculate \( w \):
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Calculate \( u + v \): \[ u + v = 250,000 , \text{km/s} + 250,000 , \text{km/s} = 500,000 , \text{km/s} \]
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Calculate \( \frac{uv}{c^2} \): \[ c^2 = (300,000 , \text{km/s})^2 = 90,000,000,000 , \text{km}^2/\text{s}^2 \] \[ uv = (250,000 , \text{km/s}) \times (250,000 , \text{km/s}) = 62,500,000,000 , \text{km}^2/\text{s}^2 \] \[ \frac{uv}{c^2} = \frac{62,500,000,000}{90,000,000,000} \approx 0.6944 \]
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Substitute these values into the velocity addition formula: \[ w = \frac{500,000 , \text{km/s}}{1 + 0.6944} = \frac{500,000}{1.6944} \approx 294,117 , \text{km/s} \]
Therefore, the speed of each particle relative to the other is approximately 294,117 km/s, which is less than 500,000 km/s and greater than 250,000 km/s.
The correct response is Between 250,000 and 500,000 km/s.
1 answer