Taking the components of a system containing multiple vectors

[Kaushik]

Please show me your work step by step as I outlined them. First step, how did you rewrite the equation
$d(h^2)=d(x_1^2)+d(c^2)$?

$\frac{d(h^2)}{dt} = \frac{d(x_1^2)}{dt} + \frac{d(c^2)}{dt}$
$2h(v_h) = 2x_1(u)$ (using chain rule)
$\frac{u}{v_h} = \frac{h}{x_1}$
Using trig,
$\frac{h}{x_1} = \frac{1}{cosθ}$
$∴ v_h = u(cosθ)$
Now, $v_h$ is the velocity along the inclined.

After this what should i do? Which triangle should i use? Please help.

Thanks.

 

[haruspex]

the string that is connected to the block is also moving with velocity u.

Each point of the string along the section from block to pulley is getting closer to the pulley at rate u. That is, of its total velocity, the component towards the pulley is u.

Each point also has a tangential component, i.e. it has some rotation about the point of contact with the pulley. The further from the pulley the greater this component.
At the point where the string touches the block, the overall velocity, v, is vertical. It therefore has a velocity v cos(θ) towards the pulley (=u) and a tangential velocity v sin(θ)=u tan(θ).