Answer
To find the magnitude of the velocity of one particle relative to the other, you can use the relativistic velocity addition formula. The formula for adding velocities relativistically is:
\[v_{\text{rel}} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}\]
Where:
- \(v_{\text{rel}}\) is the relative velocity between two objects.
- \(v_1\) and \(v_2\) are the velocities of the two objects as measured in the laboratory.
- \(c\) is the speed of light.
In this case, both particles have a velocity of \(0.9380c\) as measured in the laboratory, so \(v_1 = v_2 = 0.9380c\).
Now, plug these values into the formula:
\[v_{\text{rel}} = \frac{0.9380c + 0.9380c}{1 + \frac{(0.9380c)(0.9380c)}{c^2}}\]
Let's simplify this expression:
\[v_{\text{rel}} = \frac{1.8760c}{1 + 0.8817}\]
Now, calculate the value of \(v_{\text{rel}}\):
\[v_{\text{rel}} = \frac{1.8760c}{1.8817}\]
\[v_{\text{rel}} \approx 0.9973c\]
So, the magnitude of the velocity of one particle relative to the other is approximately \(0.9973c\).
Work Step by Step
To find the magnitude of the velocity of one particle relative to the other, you can use the relativistic velocity addition formula. The formula for adding velocities relativistically is:
\[v_{\text{rel}} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}\]
Where:
- \(v_{\text{rel}}\) is the relative velocity between two objects.
- \(v_1\) and \(v_2\) are the velocities of the two objects as measured in the laboratory.
- \(c\) is the speed of light.
In this case, both particles have a velocity of \(0.9380c\) as measured in the laboratory, so \(v_1 = v_2 = 0.9380c\).
Now, plug these values into the formula:
\[v_{\text{rel}} = \frac{0.9380c + 0.9380c}{1 + \frac{(0.9380c)(0.9380c)}{c^2}}\]
Let's simplify this expression:
\[v_{\text{rel}} = \frac{1.8760c}{1 + 0.8817}\]
Now, calculate the value of \(v_{\text{rel}}\):
\[v_{\text{rel}} = \frac{1.8760c}{1.8817}\]
\[v_{\text{rel}} \approx 0.9973c\]
So, the magnitude of the velocity of one particle relative to the other is approximately \(0.9973c\).
