Question Bank Directory

A particle moves in the x-y plane according to rule $x = a \sin \omega t$ and $y = a \cos \omega t$. The particle follows:

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is:

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?

Consider a drop of rainwater having a mass of $1\text{ g}$ falling from a height of $1\text{ km}$. It hits the ground with a speed of $50\text{ m/s}$. Take $g$ as constant with a value $10\text{ m/s}^2$. The work done by the (i) gravitational force and the (ii) resistive force of air is:

The number of possible natural oscillations of the air column in a pipe closed at one end of a length of $85 \text{ cm}$ whose frequencies lie below $1250 \text{ Hz}$ is: (velocity of sound $340 \text{ ms}^{-1}$)

The position vector of a particle $\vec{R}$ as a function of time $t$ is given by: $\vec{R} = 4\sin(2\pi t)\hat{i} + 4\cos(2\pi t)\hat{j}$, where $R$ is in meters, $t$ is in seconds and $\hat{i}, \hat{j}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is **wrong** for the motion of the particle?

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 body of mass $4m$ is lying in the $xy$-plane at rest. It suddenly explodes into three pieces. Two pieces each of mass $m$ move perpendicular to each other with equal speeds $v$. The total kinetic energy generated due to the explosion is:

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

The wave described by $y = 0.25\sin(10\pi x - 2\pi t)$, where $x$ and $y$ are in metres and $t$ in seconds, is a wave traveling along the:

If a body is moving in a circle of radius $r$ with a constant speed $v$, its angular velocity is:

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

In the hydrocarbon $\overset{6}{C}H_3-\overset{5}{C}H=\overset{4}{C}H-\overset{3}{C}H_2-\overset{2}{C}\equiv\overset{1}{C}H$, the state of hybridisation of carbons 1, 3 and 5 are in the following sequence:

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:

A car turns at a constant speed on a circular track of radius $100 \text{ m}$, taking $62.8 \text{ s}$ for every circular lap. The average velocity and average speed for each circular lap, respectively, is:

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:

A string is stretched between fixed points separated by $75.0 \text{ cm}$. It is observed to have resonant frequencies of $420 \text{ Hz}$ and $315 \text{ Hz}$. There are no other resonant frequencies between these two. The lowest resonant frequency for this string is:

A source of sound S emitting waves of frequency $100 \text{ Hz}$ and an observer O are located at some distance from each other. The source is moving with a speed of $19.4 \text{ m/s}$ at an angle of $60^{\circ}$ with the source-observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer (velocity of sound in air $330 \text{ m/s}$), is:

Two periodic waves of intensities $I_1$ and $I_2$ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is:

The dimensions $[MLT^{-2} A^{-2}]$ belong to the