For a projectile projected at angles $(45^{\circ}-\theta)$ and $(45^{\circ}+\theta)$, the horizontal ranges described by the projectile are in the ratio of:

An object is thrown along a direction inclined at an angle of $45^\circ$ with the horizontal direction. The horizontal range of the particle is equal to:

A particle moves so that its position vector is given by $\vec{r} = \cos(\omega t)\hat{x} + \sin(\omega t)\hat{y}$, where $\omega$ is a constant. Which of the following is true?

A body is moving with velocity $30 \text{ m/s}$ towards east. After $10 \text{ s}$ its velocity becomes $40 \text{ m/s}$ towards north. The average acceleration of the body is:

A ball is projected with kinetic energy $E$ at an angle of $45^\circ$ to the horizontal. At the highest point during its flight, its kinetic energy will be:

Two particles $A$ and $B$, move with constant velocities $\vec{v}_1$ and $\vec{v}_2$ respectively. At the initial moment, their position vectors are $\vec{r}_1$ and $\vec{r}_2$ respectively. The condition for particles $A$ and $B$ for their collision will be:

A particle starting from the origin $(0,0)$ moves in a straight line in the $(x,y)$ plane. Its coordinates at a later time are $(\sqrt{3}, 3)$. The path of the particle makes an angle of __________ with the $x$-axis:

If a particle moves in a circle describing equal angles in equal times, its velocity vector:

Two bodies are projected with the same velocity. If one is projected at an angle of $30^{\circ}$ and the other at an angle of $60^{\circ}$ to the horizontal, the ratio of the maximum heights reached is: