With what speed must a muon travel so that its correct (relativistic) momentum is 8.0 percent greater than the value that would be computed nonrelativistically? Give your answer as a ratio to the speed of light
To determine the speed at which a muon must travel in order to have a correct relativistic momentum that is 8.0 percent greater than its nonrelativistic momentum, we can use the following formula for momentum in relativity:
p = γ * m * v,
where p is the momentum, γ is the Lorentz factor (γ = 1 / sqrt(1 - v^2 / c^2)), m is the rest mass of the muon, v is its velocity, and c is the speed of light.
The nonrelativistic momentum can be calculated using the classical formula:
p_nonrel = m * v.
Now, let's denote the relativistic momentum as p_rel. According to the problem, p_rel should be 8.0 percent greater than p_nonrel. Mathematically, we have:
p_rel = p_nonrel + (8.0/100) * p_nonrel.
Substituting the expressions for p_rel and p_nonrel, we get:
γ * m * v = m * v + (8.0/100) * m * v.
We can simplify the equation by canceling out the mass factor:
γ * v = v + (8.0/100) * v.
Dividing both sides of the equation by v and rearranging the terms, we obtain:
γ - 1 = 0.08.
Now, we can solve for γ:
γ = 1 + 0.08.
γ = 1.08.
Finally, substituting γ into the Lorentz factor formula:
γ = 1 / sqrt(1 - v^2 / c^2),
we can solve for v / c:
1.08 = 1 / sqrt(1 - (v / c)^2).
Squaring both sides and rearranging the equation yields:
(v / c)^2 = 1 - (1 / 1.08)^2.
(v / c)^2 = 1 - 0.91975309.
(v / c)^2 = 0.08024691.
Taking the square root of both sides, we obtain:
v / c = sqrt(0.08024691).
v / c = 0.283088.
Therefore, the ratio of the muon's speed to the speed of light is approximately 0.283088.