Particle, kinetic energy, decay point

[Yroyathon]

Homework Statement

The lifetime of a particle is 1.0*10^-16 s in the particle's rest frame. With what energy would one of these particles have to be produced so that its decay point is distinguishable from its production point in a photographic plate? Assume that a 1 mm separation is required for a measurement. The particle mass corresponds to mc^2 = 150 MeV.

Homework Equations

energy of particle with no potential energy
E = gamma * m * c^2
where gamma = 1 / sqrt(1 - (v^2/c^2))

kinetic energy of a particle
K = m * c^2 * (gamma - 1)

The Attempt at a Solution

I tried calculating the velocity, v= 1*10^(-3)m / 1 * 10^(-16) s = 1 * 10^13 m/s. but this velocity is greater than c, which is both bad/impossible and prevents me from using other equations I have.

i feel like first I need to resolve this velocity problem before I can continue, since most of my energy equations in the textbook involve velocity, and this problem involving both a time and a distance lead me to believe velocity will be involved.

suggestions? hints?... anything would be appreciated.

Thanks.
,Yroyathon

 

[rl.bhat]

Use uncertainty principle.

 

[Yroyathon]

well, Hm.

the uncertainty principle is in the next chapter, we haven't gone over it yet. So either there's been some sort of screwup and the prof or anyone hasn't noticed it, or there has to be a way to solve it without the uncertainty principle.

 

[Nabeshin]

Remember, the lifetime given is that in the particle's rest frame. The lifetime seen in the lab's reference frame will be different.

 

[Yroyathon]

aha. yes. that's definitely in this chapter, thanks. i'll dig back into the problem now and try to incorporate that info.

 

[Yroyathon]

ok then. so I tried using the Lorentz transformations we have for t -> t', and x -> x', but I didn't now what to put for the u, the speed of the inertial frame. So I've got u's and gamma's floating around everywhere gumming up the works, keeping me from being able to solve explicitly for t and x', the time in the observation frame and the displacement in the particle's frame. Maybe this isn't the correct approach.

We've got some momentum and energy in special relativity material/equations, but my physic1&2 is pretty rusty, so setting up the problem is difficult. could someone explain at least part of the process or ideas involved?

I'm just flailing at the moment because I don't really know which approach to use now.