$A$ parachutist of weight $w$ strikes the ground with his legs fixed and comes to rest with an upward acceleration of magnitude $3g$. The force exerted on him by the ground during landing is:

In the figures $(i), (ii)$ and $(iii)$ shown, the objects $A, B$ and $C$ have the same mass $m$. The string, spring, and pulley are massless. Object $C$ strikes object $B$ with velocity $u$ in each case and sticks to it. Find the ratio of the velocity of $B$ in case $(i)$ to $(ii)$ to $(iii)$.

Assertion $(A):$ Two identical trains move in opposite senses in the equatorial plane with the same speeds relative to the Earth's surface. They have equal magnitude of normal reaction.
Reason $(R):$ The trains have different centripetal accelerations due to different speeds.

At time $t=0$,a force $F=\alpha t$,where $t$ is time in seconds,is applied to a body of mass $1 \text{ kg}$,resting on a smooth horizontal plane. If the direction of the force makes an angle of $45^{\circ}$ with the horizontal,then the velocity of the body at the moment of its breaking off the plane is

$A$ block is placed on a rough horizontal plane. $A$ time-dependent horizontal force $F = kt$ acts on the block,where $k$ is a positive constant. The acceleration-time graph of the block is

An infinite number of masses are placed on a frictionless table and they are connected via massless strings. Their masses follow the sequence $m, \frac{m}{2}, \frac{m}{6}, \ldots, \frac{m}{n!}, \ldots$ and they are further connected to a mass $m$ that hangs over a massless pulley. The acceleration of the hanging mass is