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 × 108 m/s, m proton = 1.67 × 10-27 kg): *
To find the speed of the proton in the lab, we need to use the concept of relativistic momentum and the equation relating momentum and energy.
The relativistic momentum (p) of an object is given by the equation:
p = γ * m * v
where γ is the Lorentz factor, m is the mass of the object, and v is its velocity.
The Lorentz factor (γ) is given by:
γ = 1 / √(1 - v^2/c^2)
where c is the speed of light.
We are given the momentum (p) of the proton as 1.00 × 10^(-19) kg · m/s.
We are also given the rest energy (E) of the proton as 0.150 nJ.
The rest energy of an object is related to its mass (m) by the equation:
E = m * c^2
where c is the speed of light.
We are given the mass of the proton as 1.67 × 10^(-27) kg.
First, let's find the velocity of the proton using the equation for relativistic momentum:
p = γ * m * v
Solving for v, we get:
v = p / (γ * m)
Now, let's find the Lorentz factor (γ) using the equation for rest energy:
E = m * c^2
Solving for γ, we get:
γ = E / (m * c^2)
Now, let's substitute the given values into the equations:
m = 1.67 × 10^(-27) kg
p = 1.00 × 10^(-19) kg · m/s
E = 0.150 nJ = 0.150 × 10^(-9) J
c = 3.00 × 10^8 m/s
Substituting these values into the equations, we can find the velocity of the proton in the lab.