A New Look at Momentum — Question 4

Question 4

Parallel and transverse forces produce different acceleration responses

At one instant a particle has velocity $\mathbf v$ and acceleration $\mathbf a$. Differentiate $\mathbf p=\gamma m\mathbf v$, using $d\gamma/dt=\gamma^3\mathbf v\cdot\mathbf a/c^2$.

1. Derive the force components parallel and perpendicular to the velocity.

2. At $v=0.80c$, compare the acceleration magnitudes caused by equal-magnitude purely parallel and purely transverse forces.

Question 4 - Solution

Parallel and transverse forces produce different acceleration responses

(a) Method and calculation

$$\mathbf F=\gamma m\mathbf a+\gamma^3m\frac{\mathbf v\cdot\mathbf a}{c^2}\mathbf v.$$$$\boxed{F_\parallel=\gamma^3ma_\parallel},\qquad\boxed{F_\perp=\gamma ma_\perp}.$$

(b) Interpretation and further calculation

$$|F_\parallel|=|F_\perp|:\quad\frac{|a_\perp|}{|a_\parallel|}=\gamma^2.$$$$\boxed{|a_\perp|/|a_\parallel|=25/9}.$$

Physical check

The invariant mass is unchanged. The different responses arise from differentiating a velocity-dependent momentum.