If a particle has kinetic energy K = 60 MeV when its momentum magnitude p = 100 MeV/c, what is the ratio m/me of its mass to the electron mass?
161.079
To find the ratio m/me of the particle's mass to the electron mass, we can use the relativistic expression for kinetic energy and momentum:
K = [(γ - 1) * m * c²]
p = γ * m * v
Where:
K = Kinetic energy of the particle
p = Magnitude of momentum
m = Mass of the particle
me = Mass of the electron
c = Speed of light
v = Velocity of the particle
γ = Lorentz factor (γ = 1 / sqrt(1 - (v²/c²)))
We are given:
K = 60 MeV
p = 100 MeV/c
From the given information, we can find the Lorentz factor γ:
γ = p / (m * v)
To simplify the calculation, we can plug in the values p = 100 MeV/c and γ into the expression for γ:
γ = 100 MeV/c / (m * v)
Now, we can substitute the expression for γ into the expression for kinetic energy:
K = [(γ - 1) * m * c²]
Substituting the expression for γ:
60 MeV = [((100 MeV/c) / (m * v)) - 1] * m * c²
Simplifying the equation further:
60 MeV = (100 MeV/c) / (m * v) - m * c²
To proceed further, we need to ensure that the units match up. As we know, c = 3 x 10⁸ m/s, so we need to convert the units:
60 MeV = (100 MeV / (m * v)) - (m * (3 x 10⁸)² m²/s²)
Now, let's rearrange the equation to isolate the mass ratio m/me:
60 MeV = 100 MeV / (m * v) - (m * 9 x 10¹⁶) kg
Divide both sides of the equation by 60 MeV:
1 = (100 MeV / (60 MeV)) / (m * v) - (m * 9 x 10¹⁶) kg
Simplifying the expression:
1 = (5/3) / (m * v) - (m * 9 x 10¹⁶) kg
Multiply both sides by (m * v):
(m * v) = (5/3) - (m * 9 x 10¹⁶) kg
Now, divide both sides by [(5/3) - (m * 9 x 10¹⁶) kg]:
(m * v) / [(5/3) - (m * 9 x 10¹⁶) kg] = 1
Finally, we have the ratio m/me:
m / (me) = w = (m * v) / [(5/3) - (m * 9 x 10¹⁶) kg]
Now, we can solve for the value of m / me by substituting the given values for K and p and then finding the appropriate value for v.