In a particle accelerator, a proton is accelerated to 0.9999 times the speed of light. Calculate the relativistic mass of the proton and the energy required to accelerate it.
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To calculate the relativistic mass of the proton, we can use the relativistic mass formula:
\(m = \gamma \cdot m_0\)
where:
\(m\) = relativistic mass,
\(m_0\) = rest mass (mass at rest),
\(\gamma\) = Lorentz factor.
The Lorentz factor can be calculated using the formula:
\(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)
where:
\(v\) = velocity of the proton,
\(c\) = speed of light.
Given that the proton is accelerated to 0.9999 times the speed of light, \(v = 0.9999c\).
Plugging this value into the equation, we have:
\(\gamma = \frac{1}{\sqrt{1 - (0.9999c)^2/c^2}}\)
\(\gamma = \frac{1}{\sqrt{1 - 0.9999^2}}\)
\(\gamma \approx 70.7\)
Next, we need to know the rest mass of a proton, which is approximately \(m_0 = 1.67 \times 10^{-27}\) kg.
Finally, we can calculate the relativistic mass:
\(m = \gamma \cdot m_0\)
\(m \approx 70.7 \cdot 1.67 \times 10^{-27}\) kg
\(m \approx 1.18 \times 10^{-25}\) kg
So, the relativistic mass of the proton is approximately \(1.18 \times 10^{-25}\) kg.
To calculate the energy required to accelerate the proton, we can use the formula:
\(E = \gamma \cdot m_0 \cdot c^2\)
Plugging in the values, we have:
\(E = 70.7 \cdot 1.67 \times 10^{-27} \cdot (3 \times 10^8)^2\) J
\(E \approx 2.51 \times 10^{-10}\) J
So, the energy required to accelerate the proton is approximately \(2.51 \times 10^{-10}\) J.
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