In the lab, a relativistic proton has a momentum of 1.00 ×10^-19 kg ∙ m/s and a rest energy of 0.150 nJ. What is the speed
of the proton in the lab? (c = 3.00 × 10^8 m/s, mass of proton = 1.67 ×10^-27 kg)
To find the speed of the proton in the lab, we can use the relativistic equation for momentum.
The relativistic equation for momentum is given by:
p = γ * m * v
where,
p is the momentum of the particle,
γ is the Lorentz factor,
m is the rest mass of the particle,
v is the velocity of the particle.
In this case, the momentum of the proton is given as 1.00 × 10^-19 kg ∙ m/s and the rest energy is given as 0.150 nJ. We can find the velocity of the proton by solving for v in the relativistic equation for momentum.
First, we need to calculate the relativistic mass (m_rel) of the proton using the rest energy (E_rest) and the speed of light (c).
The relativistic mass (m_rel) is given by:
m_rel = E_rest / (c^2)
m_rel = 0.150 nJ / (3.00 × 10^8 m/s)^2
Next, we can rearrange the relativistic equation for momentum to solve for velocity (v):
v = p / (γ * m)
To find γ, we can use the formula:
γ = 1 / sqrt(1 - (v^2 / c^2))
Rearranging the equation for γ, we get:
γ^2 = 1 / (1 - (v^2 / c^2))
Now, let's plug in the given values:
m = 1.67 × 10^-27 kg (mass of proton)
c = 3.00 × 10^8 m/s (speed of light)
p = 1.00 × 10^-19 kg.m/s (momentum of the proton)
E_rest = 0.150 nJ (rest energy of the proton)
1. Convert the rest energy E_rest to joules (J):
E_rest = 0.150 nJ = 0.150 × 10^-9 J
2. Calculate the relativistic mass (m_rel):
m_rel = E_rest / (c^2) = (0.150 × 10^-9 J) / ((3.00 × 10^8 m/s)^2)
3. Calculate γ:
γ = 1 / sqrt(1 - (v^2 / c^2))
4. Rearrange γ^2 = 1 / (1 - (v^2 / c^2)) to solve for γ.
5. Plug in the values of p, γ, and m into the velocity equation:
v = p / (γ * m)
6. Substitute the given values of p, γ, and m_rel to calculate the velocity (v) of the proton in the lab.