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  4. Two cars A and B start moving from the same point with same velocity v=5 km / minute. Car A moves towards north and car B is moving towards east. What is the relative velocity of B with respect to A ?

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Q. Two cars $A$ and $B$ start moving from the same point with same velocity $v=5 km /$ minute. Car $A$ moves towards north and car $B$ is moving towards east. What is the relative velocity of $B$ with respect to $A$ ?

Motion in a Plane

Solution:

The velocity of car $A$ and car $B$ are represented by the vectors as shown in the figure.
image
Given : $v_{A}=v_{B}=v=5 km /$ minute
The relative velocity of car $B$ w.r.t to car $A$ is given by
$\vec{v}_{B A} =\vec{v}_{B}-\vec{v}_{A} $
$\left|\vec{v}_{B A}\right| =\sqrt{v_{B}^{2}+\left(-v_{A}\right)^{2}+2\left(v_{B}\right)\left(-v_{A}\right) \cos \left(90^{\circ}\right)} $
$ =\sqrt{(5)^{2}+(-5)^{2}+(0)}=5 \sqrt{2} km $ minute
The direction $\beta$ is given by
$\tan \beta=\frac{v_{B}}{v_{A}}=\frac{5}{5}=1 $
or $ \beta=45^{\circ}$ in south-east