$A$ particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\vec{v} = \alpha(y \hat{i} + 2x \hat{j})$,where $\alpha$ is a non-zero constant. What is the force $\vec{F}$ acting on the particle?

In the following cases,identify the source providing the centripetal force:
$(i)$ Motion of the Earth around the Sun.
$(ii)$ Motion of an electron around the nucleus.
$(iii)$ Motion of a vehicle on a horizontal curved road.

$A$ smooth circular groove has a smooth vertical wall as shown in the figure. $A$ block of mass $m$ moves against the wall with a speed $v$. Which of the following curves represents the correct relation between the normal reaction $(N)$ on the block by the wall and the speed of the block $(v)$?

$A$ particle is moving eastwards with a velocity of $5\,m/s$. In $10\,s$,the velocity changes to $5\,m/s$ northwards. The average acceleration in this time is

$A$ particle,moving with uniform speed $v$,changes its direction by angle $\theta$ in time $t$. The magnitude of its average acceleration during this time is:

$A$ hill is $500 \, m$ high. Supplies are to be sent across the hill,using a cannon that can hurl packets at a speed of $125 \, m/s$ over the hill. The cannon is located at a distance of $800 \, m$ from the foot of the hill and can be moved on the ground at a speed of $2 \, m/s$; so that its distance from the hill can be adjusted. What is the shortest time in which a packet can reach the ground on the other side of the hill? Take $g = 10 \, m/s^2$.