Question

A block is dragged on a smooth plane with the help of a rope which moves with a velocity $v$ as shown in the figure. The horizontal velocity of the block is

• A. $\frac{v}{\sin \theta }$
• B. $v\sin \theta$
• C. $\frac{v}{\cos \theta }$
• D. $v\cos \theta$

Answer 1

Answer: (A)

Let at any instant of time, the length $AB$ be $l$.
Here angle $\theta$ and length $l$ vary with time, then using Pythagorus theorem in $△ABC$

Differentiating both sides w.r.t. $t$, we get
$x^2+y^2=l^2$
Differentiating both sides w.r.t. '$t$', we get
or $2x\frac{dx}{dt}+2y\frac{dy}{dt}=2l\frac{dl}{dt}$
As, there no vertical motion of the block, so
$\frac{dy}{dt}=0,\frac{dx}{dt}=v_x$ and $\frac{dl}{dt}=v$
$\therefore 2x{v}_x=2lv$
or $v_x=\frac{l}{x}v$
or $v_x=\frac{v}{\left(\frac{x}{l}\right)}=\frac{v}{\sin \theta }$