A particle has two velocities of equal magnitude inclined to each other at an angle .

Question

A particle has two velocities of equal magnitude inclined to each other at an angle $	heta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $	heta$ is:

Accepted Answer

Let both velocities have magnitude $u$. The original resultant $R=2u\cos\dfrac\theta2$ bisects the angle $\theta$ between them, so it makes angle $\dfrac\theta2$ with each. Halving one velocity and applying the angle bisector condition (via the angle bisector theorem in the relevant triangle) gives $R'=u$, and:

$$2u\cos\frac\theta2 = u \implies \cos\frac\theta2=\frac12=\cos60^{\circ} \implies \theta=120^{\circ}$$

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