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Question Number 19362 by Tinkutara last updated on 10/Aug/17

The block Q moves to the right with a constant velocity v_0 as shown in figure. The relative velocity of body P with respect to Q is (assume all pulleys and strings are ideal)

$$\mathrm{The}\:\mathrm{block}\:{Q}\:\mathrm{moves}\:\mathrm{to}\:\mathrm{the}\:\mathrm{right}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{constant}\:\mathrm{velocity}\:{v}_{\mathrm{0}} \:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}. \\ $$$$\mathrm{The}\:\mathrm{relative}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{body}\:{P}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:{Q}\:\mathrm{is}\:\left(\mathrm{assume}\:\mathrm{all}\:\mathrm{pulleys}\:\mathrm{and}\right. \\ $$$$\left.\mathrm{strings}\:\mathrm{are}\:\mathrm{ideal}\right) \\ $$

Commented by Tinkutara last updated on 10/Aug/17

Commented by ajfour last updated on 10/Aug/17

2x_P =3x_Q ⇒ 2v_P =3v_Q =3v_0 v_P =(3/2)v_0 ⇒ v_P −v_Q =(3/2)v_0 −v_0 v_(rel) =(v_0 /2) .

$$\mathrm{2x}_{\mathrm{P}} =\mathrm{3x}_{\mathrm{Q}} \\ $$$$\Rightarrow\:\:\mathrm{2v}_{\mathrm{P}} =\mathrm{3v}_{\mathrm{Q}} =\mathrm{3v}_{\mathrm{0}} \\ $$$$\:\:\:\:\:\mathrm{v}_{\mathrm{P}} =\frac{\mathrm{3}}{\mathrm{2}}\mathrm{v}_{\mathrm{0}} \:\:\:\Rightarrow\:\:\mathrm{v}_{\mathrm{P}} −\mathrm{v}_{\mathrm{Q}} =\frac{\mathrm{3}}{\mathrm{2}}\mathrm{v}_{\mathrm{0}} −\mathrm{v}_{\mathrm{0}} \\ $$$$\:\:\:\mathrm{v}_{\mathrm{rel}} =\frac{\mathrm{v}_{\mathrm{0}} }{\mathrm{2}}\:. \\ $$

Commented by Tinkutara last updated on 11/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

Commented by ajfour last updated on 10/Aug/17

towards Q.

$$\mathrm{towards}\:\mathrm{Q}. \\ $$

Commented by ajfour last updated on 11/Aug/17

let +ve direction is towards right. v_Q =v_0 , v_P =(3/2)v_0 v_(rel) of P with respect to Q is = v_P −v_Q =(3/2)v_0 −v_0 =(v_0 /2) (+ve) hence towards right means towards Q.

$$\mathrm{let}\:+\mathrm{ve}\:\mathrm{direction}\:\mathrm{is}\:\mathrm{towards}\:\mathrm{right}. \\ $$$$\mathrm{v}_{\mathrm{Q}} =\mathrm{v}_{\mathrm{0}} \:\:,\:\mathrm{v}_{\mathrm{P}} =\frac{\mathrm{3}}{\mathrm{2}}\mathrm{v}_{\mathrm{0}} \\ $$$$\mathrm{v}_{\mathrm{rel}} \:\mathrm{of}\:\mathrm{P}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{Q}\:\:\mathrm{is}\:=\:\mathrm{v}_{\mathrm{P}} −\mathrm{v}_{\mathrm{Q}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{v}_{\mathrm{0}} −\mathrm{v}_{\mathrm{0}} =\frac{\mathrm{v}_{\mathrm{0}} }{\mathrm{2}}\:\:\:\:\left(+\mathrm{ve}\right) \\ $$$$\mathrm{hence}\:\mathrm{towards}\:\mathrm{right}\:\mathrm{means}\:\mathrm{towards}\:\mathrm{Q}. \\ $$

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