Kinematics

The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :

The position vector $\vec r$ of a particle of mass $m$ is given by the following equation
$$\vec r(t)=\alpha t^3\,\hat i+\beta t^2\,\hat j,$$
where $\alpha=10/3\ \mathrm{m\,s^{-3}}$, $\beta=5\ \mathrm{m\,s^{-2}}$ and $m=0.1$ kg. At $t=1$ s, which of the following statement(s) is(are) true about the particle?

The distance $r$ of the block at time $t$ is

Which of the following option correctly describes the variation of the speed $v$ and acceleration '$a$' of a point mass falling vertically in a viscous medium that applies a force F = −kv, where 'k' is a constant, on the body ? (Graphs are schematic and not drawn to scale)

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

Velocity-time graph for a body of mass 10 kg is shown in figure. Work-done on the body in first two seconds of the motion is :

Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first ?

(Assume stones do not rebound after hitting the ground and neglect air resistance, take $g=10$ m/s$^2$)

(The figures are schematic and not drawn to scale)

$x$ and $y$ displacements of a particle are given as $x(t)=a\sin\omega t$ and $y(t)=a\sin2\omega t$. Its trajectory will look like :

If a body moving in a circular path maintains constant speed of 10 ms$^{-1}$, then which of the following correctly describes relation between acceleration and radius ?

A vector $\vec A$ is rotated by a small angle Δθ radians (Δθ<<1) to get a new vector $\vec B$. In that case $|\vec B-\vec A|$ is :

Airplanes A and B are flying with constant velocity in the same vertical plane at angles 30° and 60° with respect to the horizontal respectively as shown in figure. The speed of A is $100\sqrt3\ \mathrm{m\,s^{-1}}$. At time $t=0$ s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $t=t_0$, A just escapes being hit by B, $t_0$ in seconds is

A rocket is moving in a gravity free space with a constant acceleration of $2\ \mathrm{m\,s^{-2}}$ along $+x$ direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in $+x$ direction with a speed of $0.3\ \mathrm{m\,s^{-1}}$ relative to the rocket. At the same time, another ball is thrown in $-x$ direction with a speed of $0.2\ \mathrm{m\,s^{-1}}$ from its right end relative to the rocket. The time in seconds when the two balls hit each other is

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $K$ with time $t$ most appropriately? The figures are only illustrative and not to the scale.

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the list.

From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken by the particle, to hit the ground, is $n$ times that taken by it to reach the highest point of its path.

The relation between $H$, $u$ and $n$ is :

The position of a projectile launched from the origin at $t=0$ is given by $\vec r=\left(40\hat i+50\hat j\right)$ m at $t=2$ s. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is (take $g=10$ ms$^{-2}$).

The initial speed of a bullet fired from a rifle is 630 m/s. The rifle is fired at the centre of a target 700 m away at the same level as the target. How far above the centre of the target the rifle must be aimed in order to hit the target ?

A body of mass 5 kg under the action of constant force $\vec F=F_x\hat i+F_y\hat j$ has velocity at $t=0$ s as $\vec v=\left(6\hat i-2\hat j\right)$ m/s and at $t=10$ s as $\vec v=+6\hat j$ m/s. The force $\vec F$ is :

A person climbs up a stalled escalator in 60 s. If standing on the same but escalator running with constant velocity he takes 40 s. How much time is taken by the person to walk up the moving escalator ?

A particle is moving in a circular path of radius a, with a constant velocity $v$ as shown in the figure. The center of circle is marked by 'C'. The angular momentum from the origin O can be written as :