Relative velocity

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds 100 km/h and 80 km/h, respectively such that car $A$ is moving ahead of car $B$. A person in car $B$ throws a stone with a speed $v$ so that it hits the car $A$ with a speed of 5 m/s. The value of $v$ is ________ km/h.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1 = 1.21\,r_2$. The time period of the second satellite as measured from the geostationary satellite is $\dfrac{24}{p}$ hours. The value of $p$ is _____.

A ball is thrown from the location $(x_0, y_0) = (0, 0)$ of a horizontal playground with an initial speed $v_0$ at an angle $\theta_0$ from the $+x$-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $(x_1, y_1) = (L, 0)$. The stone is thrown at an angle $(180 - \theta_1)$ from the $+x$-direction with a suitable initial speed. For a fixed $v_0$, when $(\theta_0, \theta_1) = (45^{\circ}, 45^{\circ})$, the stone hits the ball after time $T_1$, and when $(\theta_0, \theta_1) = (60^{\circ}, 30^{\circ})$, it hits the ball after time $T_2$. In such a case, $(T_1/T_2)^2$ is _____.

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60\ \mathrm{cm\,s^{-1}}$, the speed of the tip of the person's shadow on the ground with respect to the person is _____ $\mathrm{cm\,s^{-1}}$.

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $\mathrm{m\,s^{-1}}$) at time $t = \dfrac\pi3$ s.

Which one of the following options is correct?

An object and a concave mirror of focal length $f = 10$ cm both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_0 = 15\ \mathrm{cm\,s^{-1}}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$. When $u = 30$ cm, the speed of the mirror $V_m$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_m$ is _____ $\mathrm{cm\,s^{-1}}$.

A fighter jet is flying horizontally at a certain altitude with a speed of 200 m s$^{-1}$. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $\theta$ with the horizontal, to hit the jet. If the bullet speed is 400 m/s, the value of $\theta$ will be ______°.

A girl standing on road holds her umbrella at 45° with the vertical to keep the rain away. If she starts running without umbrella with a speed of $15\sqrt2$ kmh$^{-1}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

At $t=0$, truck, starting from rest, moves in the positive $x$-direction at uniform acceleration of 5 ms$^{-2}$. At $t=20$ s, a ball is released from the top of the truck. The ball strikes the ground in 1 s after the release. The velocity of the ball, when it strikes the ground, will be :

(Given $g=10$ ms$^{-2}$)

A bag is gently dropped on a conveyor belt moving at a speed of 2 m/s. The coefficient of friction between the conveyor belt and bag is 0.4. Initially, the bag slips on the belt before it stops due to friction. The distance travelled by the bag on the belt during slipping motion, is : [Take g = 10 m/s$^{-2}$]

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 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 :

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)

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 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 ?

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y$-$z$ plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed $\frac18$ rotation, (ii) their range is less than half the disc radius, and (iii) $\omega$ remains constant throughout. Then

Two identical discs of same radius $R$ are rotating about their axes in opposite directions with the same constant angular speed $\omega$. The discs are in the same horizontal plane. At time $t=0$, the points $P$ and $Q$ are facing each other as shown in the figure. The relative speed between the two points $P$ and $Q$ is $v_r$. In one time period $(T)$ of rotation of the discs, $v_r$ as a function of time is best represented by

The figure shows a system consisting of (i) a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega/2$. The ring and disc are separated by frictionless ball bearings. The system is in the $x$-$z$ plane. The point $P$ on the inner disc is at a distance $R$ from the origin, where $OP$ makes an angle of 30° with the horizontal. Then with respect to the horizontal surface,

STATEMENT-1: For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.

and

STATEMENT-2: If the observer and the object are moving at velocities $\vec V_1$ and $\vec V_2$ respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is $\vec V_2-\vec V_1$.

The speed of sound of the whistle is