# Trajectory in an arbitrary electrostatic field

**Electricity and Magnetism**Level 4

A charged particle of mass m and charge q moves in an electrostatic field created by an unknown system of static charges. The particle starts from point A with initial velocity is \(v_{1}=1 m/s\) and follows certain trajectory. What should be the velocity (magnitude) \(v_{2}\) **in m/s** of a particle of mass \(M=10 m\) and charge \(Q=7 q\) at point A in order to follow the same trajectory?

Hint: Compute \[\frac{d}{ds} \frac{\vec{v}}{v},\] where s is the arc length along the trajectory:

\(s=\int_0^t v(t') dt'\).

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