It is possible for some fundamental particles to “violate” conservation of energy by creating and quickly re-absorbing another particle. For example, a proton can emit a π⁺ according to p = n + π⁺, where the n represents a neutron. The π⁺ has a mass of 140 MeV/c². The re-absorption must occur within a time Δt consistent with the uncertainty principle. 

(a) Considering the example shown, by how much ΔE is energy conservation violated? (Ignore kinetic energy.)

(b) For how long Δt can the π⁺ exist?

(c) Assuming that the  π⁺ is moving at nearly the speed of light, how far from the nucleus could it get in the time Δt? (As we will discuss in Chapter 11, this is the approximate range of the strong nuclear force.) 

Assuming that as soon as one pion is re-absorbed, another is emitted, how many pions would be recorded by a “nucleon camera” with a shutter speed of 1 μs?