The engine of a jet aircraft applies a thrust force of $10^5\, N$ during take-off and causes the plane to attain a velocity of $1\, km/s$ in $10\, s$. The mass of the plane is:

$A$ force $\vec{F}=(40 \hat{i}+10 \hat{j}) \, N$ acts on a body of mass $5 \, kg$. If the body starts from rest,its position vector $\vec{r}$ at time $t=10 \, s$ will be -

$A$ force $\vec{F_1}$ of magnitude $4 \ N$ acts on an object of mass $1 \ kg$ at the origin in a direction $30^{\circ}$ above the positive $x$-axis. $A$ second force $\vec{F_2}$ of magnitude $4 \ N$ acts on the same object in the direction of the positive $y$-axis. The magnitude of the acceleration of the object is nearly: (in $m \ s^{-2}$)

The motion of a particle of mass $m$ is described by $y = ut + \frac{1}{2}gt^2$. Find the force acting on the particle.

The velocity $(v)$ of a particle (under a force $F$) depends on its distance $(x)$ from the origin (with $x > 0$) as $v \propto \frac{1}{\sqrt{x}}$. Find how the magnitude of the force $(F)$ on the particle depends on $x$.

The velocity of a body of mass $2 \, kg$ as a function of $t$ is given by $\vec{v}(t) = 2t \hat{i} + t^2 \hat{j}$. Find the momentum and the force acting on it at time $t = 2 \, s$.