Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$. If their time periods are the same,then the ratio of their angular velocities will be:

Two particles $A$ and $B$ are moving in the $XY$ plane. Particle $A$ moves along a line with equation $y = x$,while particle $B$ moves along the $X$-axis such that their $X$-coordinates are always equal. If particle $B$ moves with a uniform speed of $3 \ m/s$,what is the speed of particle $A$?

$A$ point moves in the $xy$-plane according to the following equations: $x = a \sin \omega t$ and $y = a(1 - \cos \omega t)$, where $a$ and $\omega$ are positive constants. Find the angle between the point's velocity and acceleration vectors.

The initial position of an object at rest is given by $3 \hat{i}-8 \hat{j}$. It moves with constant acceleration and reaches the position $2 \hat{i}+4 \hat{j}$ after $4 \, s$. What is its acceleration?

The average acceleration vector for a particle having a uniform circular motion over one complete revolution is

An aeroplane executes a horizontal loop at a speed of $720 \text{ km/h}$ with its wings banked at $45^{\circ}$. What is the radius of the loop (in $\text{ km}$)? Take $g = 10 \text{ m/s}^2$.