In the figure, the pulley $P$ moves to the right with a constant speed $u$. The downward speed of $A$ is $v_{\mathrm{A}}$ and the speed of $B$ to the right is $v_{\mathrm{B}}$.

 

In an imaginary atmosphere, the air exerts a small force $F$ on any particle in the direction of the particle’s motion. A particle of mass $m$ projected upward takes time $t_1$ in reaching the maximum height and $t_2$ in the return journey to the original point. Then

System shown in figure is in equilibrium and at rest. The spring and string are massless. Now the string is cut. The acceleration of mass $2m$ and $m$ just after the string is cut will be:

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance $a$ from the centre $P$ (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$ is

Two fixed frictionless inclined planes making an angle $30°$ and $60°$ with the vertical are shown in the figure. Two blocks $A$ and $B$ are placed on the two planes respectively. What is the relative vertical acceleration of $A$ with respect to $B$?

(Use $\sin 30°=\cos 60°=\frac{1}{2} \text{and} \sin 60°=\cos 30°=\frac{\sqrt{3}}{2}$)

A particle of mass $m$ is at rest at the origin at time $t=0$. It is subjected to a force $F\left(t\right)=F_0{e}^{–\mathrm{bt}}$ in the $x$-direction. Its speed $v\left(t\right)$ is depicted by which of the following curves?