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Self Induction

A particle mo...

Question

A particle moves on a given straight line with a constant speed v. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that $\to O P \times \to v$ is independent of the position P.

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Solution

The particle moves on the striaght line PP' at speed v.
From the figure,
$\to O P \times v = (O P) v sin θ^n = v (O P) sin θ^n = v (O Q)^n$
It can be seen from the figure, $O Q = O P sin θ = O P^{'} sin θ^{'}$
So, whatever may be the position of the particle, the magnitude and direction of $\to O P \times \to v$ remain constant.
$∴ \to O P \times \to v$ is independent of the position P.

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