$A$ body of mass $0.5 \,kg$ is projected under gravity with a speed of $98 \,m/s$ at an angle of $30^\circ$ with the horizontal. The change in momentum (in magnitude) of the body is ......... $N-s$.

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 $\vec{F} = -m \omega^2 \vec{r}$.

$A$ stone is projected with a velocity $20 \sqrt{2} \, m/s$ at an angle of $45^{\circ}$ to the horizontal. The average velocity of the stone during its motion from the starting point to its maximum height is $.......... \, m/s$ (take $g = 10 \, m/s^2$).

$A$ projectile is thrown with a velocity of $10\,m/s$ at an angle of $60^{\circ}$ with the horizontal. The time interval between the moments when the speed is $\sqrt{5g}\,m/s$ is $..........\,s$ (take $g=10\,m/s^2$).

$A$ boy is moving with a constant speed $v$ on a small trolley towards a distant circle as shown in the figure. $A$ point mass is moving on the circle with a constant speed $v$. What is the frequency of change in the magnitude of the relative velocity of the point mass,as observed by the boy?

If a person can throw a stone to a maximum height of $h$ meters vertically, then the maximum distance through which it can be thrown horizontally by the same person is