Kinematics

If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph ?

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)

A uniform electric field, $\vec E = -400\sqrt3\,\hat y\ \mathrm{NC^{-1}}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times10^{6}\ \mathrm{m\,s^{-1}}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take $\dfrac qm = 10^{10}\ \mathrm{C\,kg^{-1}}$. Then

Starting at time $t = 0$ from the origin with speed $1\ \mathrm{m\,s^{-1}}$, a particle follows a two-dimensional trajectory in the $x$-$y$ plane so that its coordinates are related by the equation $y = \dfrac{x^2}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_x$ and $a_y$, respectively. Then

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $u_0$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_0/\alpha$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8V_1$, the value of $\alpha$ is ______.

Consider a body of mass 1.0 kg at rest at the origin at time $t = 0$. A force $\vec F = (\alpha t\,\hat i + \beta\,\hat j)$ is applied on the body, where $\alpha = 1.0\ \mathrm{N\,s^{-1}}$ and $\beta = 1.0$ N. The torque acting on the body about the origin at time $t = 1.0$ s is $\vec\tau$. Which of the following statements is (are) true?

Two vectors $\vec A$ and $\vec B$ are defined as $\vec A = a\,\hat i$ and $\vec B = a(\cos\omega t\,\hat i + \sin\omega t\,\hat j)$, where $a$ is a constant and $\omega = \pi/6\ \mathrm{rad\,s^{-1}}$. If $|\vec A + \vec B| = \sqrt3|\vec A - \vec B|$ at time $t = \tau$ for the first time, the value of $\tau$, in seconds, is ______.

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $(2 - \sqrt3)/\sqrt{10}$ s, then the height of the top of the inclined plane, in metres, is ______. Take $g = 10\ \mathrm{m\,s^{-2}}$.

In the $xy$-plane, the region $y > 0$ has a uniform magnetic field $B_1\hat k$ and the region $y < 0$ has another uniform magnetic field $B_2\hat k$. A positively charged particle is projected from the origin along the positive $y$-axis with speed $v_0 = \pi\ \mathrm{m\,s^{-1}}$ at $t = 0$, as shown in the figure. Neglect gravity in this problem. Let $t = T$ be the time when the particle crosses the $x$-axis from below for the first time. If $B_2 = 4B_1$, the average speed of the particle, in $\mathrm{m\,s^{-1}}$, along the $x$-axis in the time interval $T$ is ______.

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $dK/dt = \gamma t$, where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m = 0.4$ kg is at rest on this surface. An impulse of 1.0 N s is applied to the block at time $t = 0$ so that it starts moving along the $x$-axis with a velocity $v(t) = v_0 e^{-t/\tau}$, where $v_0$ is a constant and $\tau = 4$ s. The displacement of the block, in metres, at $t = \tau$ is __________. Take $e^{-1} = 0.37$.

A ball is projected from the ground at an angle of $45^\circ$ with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of $30^\circ$ with the horizontal surface. The maximum height it reaches after the bounce, in metres, is ___________.

A particle, of mass $10^{-3}$ kg and charge 1.0 C, is initially at rest. At time $t = 0$, the particle comes under the influence of an electric field $\vec{E}(t) = E_0 \sin\omega t\, \hat{i}$, where $E_0 = 1.0\ \mathrm{N\,C^{-1}}$ and $\omega = 10^3\ \mathrm{rad\,s^{-1}}$. Consider the effect of only the electrical force on the particle. Then the maximum speed, in $\mathrm{m\,s^{-1}}$, attained by the particle at subsequent times is ____________.

Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho}\frac{d\rho}{dt}\right)$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to

Three vectors $\vec{P}$, $\vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the points $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P}$, $\vec{Q}$ and $\vec{S}$ is

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T = 0.01$ seconds and he measures the depth of the well to be $L = 20$ meters. Take the acceleration due to gravity $g = 10\ \mathrm{ms^{-2}}$ and the velocity of sound is $300\ \mathrm{ms^{-1}}$. Then the fractional error in the measurement, $\delta L/L$, is closest to

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time ?

Which graph corresponds to an object moving with a constant negative acceleration and a positive velocity ?

An object is dropped from a height h from the ground. Every time it hits the ground it looses 50% of its kinetic energy. The total distance covered as $t\to\infty$ is :

A car is standing 200 m behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration 2 m/s$^2$ and the car has acceleration 4 m/s$^2$. The car will catch up with the bus after a time of :