Relative Velocity

Classical Mechanics Level 2

Point A moves uniformly with velocity $v$ so that the vector $\textbf{v}$ is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity $u < v$ . At the initial moment of time $v\perp u$ and the points are separated by a distance $l$ . How soon will the points converge?

t=\frac{lv}{u^2}

t=\frac{lu}{u^2-v^2}

t=\frac{lu}{v^2}

t=\frac{lv}{v^2-u^2}

1 solution

Anant Badal
Aug 27, 2017

It is seen from the figure that the points A and B converge with velocity $v-u\cos\alpha$ , where the angle $\alpha$ varies with time.

The points merge provided the following two conditions are met: $\int_{0}^{\tau}\left(v-u\cos\alpha\right)dt=l,$ and $\int_{0}^{\tau}v\cos\alpha dt=u\tau ,$ where $\tau$ is the sought time. It follows from the two equations that $\tau = \frac{vl}{\left(v^2-u^2\right)}$

Since this is a multiple choice question, there is no need to do any complex math. We know that the result in the $u=0$ limit is $l/v$ . There is only one answer that satisfies that condition.

Laszlo Mihaly - 3 years, 9 months ago

Relative Velocity

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