$A$ particle is performing a uniform circular motion along a circle of radius $R$. In half the period of revolution,its displacement and distance covered are respectively:

Two stones of masses $m$ and $3m$ are whirled in horizontal circles, the heavier one in a radius $r/3$ and the lighter one in radius $r$. When both the stones experience same centripetal forces, the tangential speed of lighter stone is 'n' times that of the value of heavier stone. The value of 'n' is

The displacement vector of a particle of mass $m$ is given by $\vec{r}(t) = A \cos \omega t \hat{i} + B \sin \omega t \hat{j}$.
$(a)$ Show that the trajectory is an ellipse.
$(b)$ Show that $\vec{F} = -m \omega^2 \vec{r}$.

Two cars $S_1$ and $S_2$ are moving in coplanar concentric circular tracks in the opposite sense with the periods of revolution $3 \, min$ and $24 \, min$,respectively. At time $t = 0$,the cars are farthest apart. Then,the two cars will be

$A$ particle moves over a $xy$ plane with a constant acceleration $\vec{a} = (4.0 \, m \, s^{-2}) \hat{i} + (4.0 \, m \, s^{-2}) \hat{j}$. At time $t = 0$, the velocity is $\vec{v}_0 = (4.0 \, m \, s^{-1}) \hat{i}$. The speed of the particle when it is displaced by $6.0 \, m$ parallel to the $x$-axis is,

$A$ projectile is thrown at an angle $\theta$ to the horizontal from $O$ and it hits the ground at $O'$. During the entire motion (select the wrong statement):