A proton has the total energy 950 MeV. What's its speed?

Answer 1

We are going to use this formula to find the velocity:
E = m_0 c²/√(1–v²/c²)
Where E is the total energy, m_0 is the rest mass, c is the speed of light and v is the velocity.

The rest mass of a proton is 1.67 × 10^(-27) kg.
The speed of light is 3.00 × 10^8 m/s.

First, write the energy in joules:
1 MeV = 1.602 × 10^(-13) J
So:
950 MeV = 950 × 1.602 × 10^(-13) J
. . . . . . . = 1.522 × 10^(-10) J

The next step is to write the formula E = m_0 c²/√(1–v²/c²) on this form: v/c = ...

E = m_0 c²/√(1–v²/c²)
√(1–v²/c²) = m_0 c²/E
1–v²/c² = (m_0 c² / E)²
v²/c² = 1 – (m_0 c² / E)²
v/c = √(1 – (m_0 c² / E)²)

Now, put in the correct values for m_0, c and E.
v/c = √(1 – (1.67×10^(-27) × (3.00×10^8)² / (1.522×10^(-10)))²)
v/c = 0.16

So the velocity is v = 0.16 c.