If the instantaneous velocity of a particle projected as shown in the figure is given by $v = a \hat{i} + (b - ct) \hat{j}$,where $a, b$,and $c$ are positive constants,the range on the horizontal plane will be

$A$ body of mass $1 \, kg$ is projected from the ground at an angle of $30^{\circ}$ with the horizontal on level ground at a speed of $50 \, m/s$. The magnitude of the change in momentum of the body during its entire flight is ....... $kg \cdot m/s$ $(g = 10 \, m/s^2)$.

The range of a particle when launched at an angle of $15^\circ$ with the horizontal is $1.5 \, km$. What is the range of the projectile when launched at an angle of $45^\circ$ to the horizontal? (in $km$)

$A$ ball is thrown from a point with a speed $v_0$ at an elevation angle of $\theta$. From the same point and at the same instant,a person starts running with a constant speed $\frac{v_0}{2}$ to catch the ball. Will the person be able to catch the ball? If yes,what should be the angle of projection $\theta$?

$A$ cart is moving horizontally along a straight line with a constant speed of $30\,m/s$. $A$ projectile is to be fired from the moving cart in such a way that it will return to the cart (at the same point on the cart) after the cart has moved $80\,m$. At what velocity (relative to the cart) must the projectile be fired? (Take $g = 10\,m/s^2$)

$A$ projectile is fired from level ground at an angle $\theta$ above the horizontal. The elevation angle $\phi$ of the highest point as seen from the launch point is related to $\theta$ by the relation