A charged particle of unknown mass and momentum much larger than the... · PHYSICS Dataset #7202

Physics · modern-physics

Problem

A charged particle of unknown mass MM and momentum ∣p∣|\mathbf{p}| much larger than the electron mass mem_{e}, crosses a detector. At a certain depth inside the active material, an atomic electron is knocked out by the incoming particle, for which the polar angle θe\theta_{e} with respect to p\mathbf{p} and the energy EeE_{e} are measured. Show that the mass MM can be estimated from the following formula: M=∣p∣[Ee+meEe−mecos⁡2θe−1]12M=|\mathbf{p}|\left[\frac{E_{e}+m_{e}}{E_{e}-m_{e}} \cos ^{2} \theta_{e}-1\right]^{\frac{1}{2}}
Intuitive explanation

This problem uses elastic scattering kinematics to extract the unknown mass from a collision signature. When the incoming particle strikes an atomic electron, conservation of momentum and energy constrain the recoil electron's direction and energy to depend on the incident particle's mass—heavier particles transfer less energy at fixed angle. By measuring where the electron goes (θe\theta_e) and how much energy it carries (EeE_e), you can reverse-engineer MM using relativistic momentum and energy conservation, treating the electron's recoil as a window into the incident particle's inertia.

Intuitive explanation

This problem uses elastic scattering kinematics to extract the unknown mass from a collision signature. When the incoming particle strikes an atomic electron, conservation of momentum and energy constrain the recoil electron's direction and energy to depend on the incident particle's mass—heavier particles transfer less energy at fixed angle. By measuring where the electron goes (θe\theta_e) and how much energy it carries (EeE_e), you can reverse-engineer MM using relativistic momentum and energy conservation, treating the electron's recoil as a window into the incident particle's inertia.

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