A proton and an α particle enter a magnetic field in a direction perpendicular to it. If the force acting on the proton is twice that acting on the α- particle, then what is the ratio of their velocities? 

4:1

F_p = 2F_{\(\alpha \)} ⇒ F_p : F_{\(\alpha \)} = 2 : 1

F = q v B Sin \(\theta\) ⇒ F ∝ qv

F_p = qv_p ; F_{\(\alpha \)} = 4qv_{\(\alpha \)}

\(\frac{F_{p}}{F_\alpha}\) = \(\frac{qv_{p}}{4qv_{\alpha}}\) = \(\frac{1}{4}\)