an electron of mass m has a speed of v. at what speed would the ration of the magnitudes of the relativistic and non relativistic momenta be equal to 2?
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To find the speed at which the ratio of the magnitudes of the relativistic and non-relativistic momenta is equal to 2, we need to consider the formulas for both relativistic and non-relativistic momentum.
The non-relativistic momentum can be calculated using the equation:
p_nonrel = mass * velocity
The relativistic momentum, on the other hand, takes into account the relativistic effects and is given by the equation:
p_rel = (gamma * mass) * velocity
Where gamma (γ) is the Lorentz factor, defined as:
gamma = 1 / sqrt(1 - (v^2 / c^2))
Here, c is the speed of light in a vacuum.
Now, let's assume the non-relativistic momentum is P_nonrel and the relativistic momentum is P_rel. We are given that we want the ratio of the magnitude of P_rel to P_nonrel to be equal to 2:
|P_rel| / |P_nonrel| = 2
This implies:
(|gamma| * mass * velocity) / (mass * velocity) = 2
The mass and velocity cancel out:
|gamma| = 2
Using the equation for gamma:
1 / sqrt(1 - (v^2 / c^2)) = 2
Squaring both sides of the equation and rearranging, we get:
1 - (v^2 / c^2) = 1/4
Subtracting 1/4 from both sides:
-v^2 / c^2 = -3/4
Dividing by -1/4 (or multiplying by -4) on both sides:
v^2 / c^2 = 3/4
Taking the square root of both sides:
v / c = sqrt(3/4)
v = c * sqrt(3/4)
Therefore, the speed at which the ratio of the magnitudes of the relativistic and non-relativistic momenta is equal to 2 is v = c * sqrt(3/4), where c is the speed of light in a vacuum.