Question

A point has equal velocities in two given directions.If one of these velocities is halved, then the angle
which the resultant makes with the other is alsohalved. The angle between the velocities is
(1) 30°
(2) 60°
(3) 90°
(4) 120°​

Answer 1

Answer:

Option (4) 120°​

Explanation:

We know that the resultant R of two vector quantities P and Q, acting at an angle θ is given by

$R=\sqrt{P^2+Q^2+2PQ\cos\theta}$

And the angle ∅ made by resultant R frrom with P is given by

$\phi=\tan^{-1}(\frac{Q\sin\theta}{P+Q\cos\theta} )$

Let the magnitude of the velocity be v and the angle betwen them be θ

Then the angle ∅ of the resltant with v is given by

$\phi=\tan^{-1}(\frac{v\sin\theta}{v+v\cos\theta} )$

$\implies \phi=\tan^{-1}(\frac{\sin\theta}{1+\cos\theta} )$

Similarly when v is halved then the ange is also halved

$\implies \phi/2=\tan^{-1}(\frac{(v/2)\sin\theta}{v+(v/2)\cos\theta} )$

$\implies \phi/2=\tan^{-1}(\frac{\sin\theta}{2+\cos\theta} )$

Thus we get

$\tan\phi=(\frac{\sin\theta}{1+\cos\theta} )$

And $\tan\phi/2=(\frac{\sin\theta}{2+\cos\theta} )$

Checking the options one by one we get the value for option (4) as

$\tan\phi=\sqrt{3} \implies \phi = 60^\circ$

And $\tan(\phi/2)=\sqrt{3}/2 \implies \phi/2 = 30^\circ \implies \phi = 60^\circ$

Therefore (4) is the correct option

Therefore, the angle between the velocities is 120°​

Answer 2

Answer:

answer is 120°.

Explanation:

refer to above attachments.