Is distance traveled proportional to relativistic momentum?

[JuliusCSquared]

Homework Statement

Hi I've been modelling a particle traveling in a particle detector that has a momentum vector Px, Py, Pz which we've conveniently been using Pperpendicular (i.e. in the xy plane) and Pz.

I can calculate the distance traveled in the xy plane and I need to calculate the distance traveled in the z direction.

Homework Equations

[/B]
Knowing that

$$\vec{p}=\frac{m\vec{v}}{\sqrt{1-\vec{v}^2/c^2}}$$

The Attempt at a Solution

My guess was Momentum is proportional to the distance traveled which I can convince myself of in the case of classical momentum but knowing the momentum of this particle is of the order GeV/c I'm unsure whether I can actually do this? My assumption comes from being able to do this,

$$\vec{d}=\frac{m \Delta t}{\sqrt{1-\vec{v}^2/c^2}}\vec{v}$$

where I assumed there was a proportionality constant k such that

$$p_\perp = k d_\perp$$
$$p_z = k d_z$$

Therefore

$$d_z=\frac{p_z}{p_\perp}d_\perp$$

But I'm unsure if I can just throw in the gamma factor and the time interval into that constant k such that these equations are valid. Would this be correct or would I have to do it another way?

 

[mfb]

The direction of motion is the direction of the momentum vector. Everything else follows from basic geometry.