Here's how to calculate the momentum of a proton moving at relativistic speeds:
1. Relativistic Momentum:
The momentum of a particle moving at relativistic speeds (close to the speed of light) is given by:
```
p = γmv
```
Where:
* p is the momentum
* γ is the Lorentz factor (accounts for relativistic effects)
* m is the mass of the proton (1.6726 × 10^-27 kg)
* v is the velocity of the proton (0.86c)
2. Calculate the Lorentz Factor (γ):
```
γ = 1 / √(1 - (v²/c²))
```
Plug in the velocity (0.86c) and the speed of light (c):
```
γ = 1 / √(1 - (0.86c)²/c²)
γ = 1 / √(1 - 0.86²)
γ ≈ 1.98
```
3. Calculate the Momentum:
Now, plug the values of γ, m, and v into the momentum equation:
```
p = γmv
p ≈ 1.98 * (1.6726 × 10^-27 kg) * (0.86 * 3 × 10^8 m/s)
p ≈ 8.64 × 10^-19 kg m/s
```
Therefore, the momentum of a proton moving at 0.86c is approximately 8.64 × 10^-19 kg m/s.
Important Note: If the speed of the proton is actually 0.86 m/s (not 0.86c), then the momentum calculation would be much simpler, as the relativistic effects would be negligible at that speed. You would simply use the classical momentum formula:
```
p = mv
```
