$A$ parachutist with a total mass of $75 \,kg$ drops vertically onto sandy ground with a speed of $2 \,m/s$ and comes to a halt over a distance of $0.25 \,m$. The average force exerted by the ground on her is close to ............ $N$.

Write important guidance for solving problems in mechanics.

Find the accelerations $a_1, a_2$ and $a_3$ of the blocks if a $10\,N$ force $F$ is applied on the $3\,kg$ block as shown in the figure.

$A$ block of mass $M = 10\,kg$ rests on a horizontal table. The coefficient of friction between the block and the table is $\mu = 0.05.$ When hit by a bullet of mass $m = 50\,g$ moving with speed $v,$ which gets embedded in it,the block moves and comes to a stop after moving a distance of $2\,m$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H,$ then neglecting energy losses and taking $g = 10\,m/s^2,$ the value of $H$ is close to ................. $km$.

$A$ uniform cube of side $b$ and mass $M$ rests on a rough horizontal table. $A$ horizontal force $F$ is applied normal to one of the faces at a point at a height $3b/4$ above the base. What should be the coefficient of friction $\mu$ between the cube and the table so that it will tip about an edge before it starts slipping?

$A$ particle of mass $m=1 \ kg$ moves in the $xy$-plane. The force on it at time $t$ is $F(t)=[2 \sin (\alpha t) \hat{i}+3 \cos (\alpha t) \hat{j}] \ N$, where $\alpha=1 \ s^{-1}$. At time $t=0$, the particle is at rest at the origin. Calculate the magnitude of its position vector $r$ (in $m$) and velocity vector $v$ (in $m/s$) at time $t=\frac{\pi}{2} \ s$.