Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2021

Write down the expression for the momentum of a particle of rest mass mm, moving with velocity v\mathbf{v} where v=vv=|\mathbf{v}| is near the speed of light cc. Write down the corresponding 4-momentum.

Such a particle experiences a force F\mathbf{F}. Why is the following expression for the particle's acceleration,


not generally correct? Show that the force can be written as follows

F=mγ(γ2c2(va)v+a)\mathbf{F}=m \gamma\left(\frac{\gamma^{2}}{c^{2}}(\mathbf{v} \cdot \mathbf{a}) \mathbf{v}+\mathbf{a}\right)

Invert this expression to find the particle's acceleration as the sum of two vectors, one parallel to F\mathbf{F} and one parallel to v\mathbf{v}.

A particle with rest mass mm and charge qq is in the presence of a constant electric field E\mathbf{E} which exerts a force F=qE\mathbf{F}=q \mathbf{E} on the particle. If the particle is at rest at t=0t=0, its motion will be in the direction of E\mathbf{E} for t>0t>0. Determine the particle's speed for t>0t>0. How does the velocity behave as tt \rightarrow \infty ?

[Hint: You may find that trigonometric substitution is helpful in evaluating an integral.]

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