Relative Motion

A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is $0.5\hspace{0.17em}\mathrm{m}\mathrm{\hspace{0.17em}}{\mathrm{s}}^{-1}$ at an angle $120°$ with the direction of flow of water. The speed of water in stream is:

A boat is sent across a river with a velocity of $8 \mathrm{km }{\mathrm{hr}}^{-1}$. If the resultant velocity of the boat is $10 \mathrm{km} {\mathrm{hr}}^{-1}$, then the velocity of the river is

A bus is moving with a speed of  $10m{s}^{-1}$  on a straight road . A scooterist wishes to overtake the bus in 100 s. If the bus is at a distance of 1 km from the scooterist, with what speed should the scooterist chase the bus?

A train of 150 meters long is going towards north direction at a speed of $10\frac{m}{s}$ . A parrot flies at the speed of $5\frac{m}{s}$ towards south direction parallel to the railway track. The time taken by the parrot to cross the train is

On a frictionless horizontal surface, assumed to be the $x – y$ plane, a small trolley $A$ is moving along a straight line parallel to the $y$– axis (see figure) with a constant velocity of  $\left(\sqrt{3}-1\right)m{s}^{-1}$ . At a particular instant, when the line $OA$ makes an angle of  $45°$  with the $x$-axis, a ball is thrown along the surface from the origin $O$. Its velocity makes an angle $\varphi$ with the $x$-axis and it hits the trolley.​
The motion of the ball is observed from the frame of the trolley and the velocity vector of the ball makes an angle $\varphi$ with the x-axis in this frame.


Find the speed of the ball with respect to the surface if  $\varphi =\frac{4\theta }{3}.$

A ball is thrown from a point with a speed $v_0$  at an angle of projection $\theta$ . from the same point and at the same instant, a person starts running with a constant speed  $\frac{v_0}{2}$ to catch the ball. will the person be able to catch the ball? If yes, what should be the angle of projection?

Four persons $K$, $L$, $M$, $N$ are initially at the four corners of a square of side $d$. Each person now moves with a uniform speed  $v$ in such a way that $K$ always moves directly towards $L$, L directly towards $M$, $M$ directly towards $N$, and $N$ directly towards $K$. The four persons will meet at a time

A jet aeroplane traveling from east to west, at a speed of $500 \mathrm{km} {\mathrm{h}}^{-1}$, ejects out gases of combustion at a speed of $1500 \mathrm{km} {\mathrm{h}}^{-1}$, with respect to the jet plane. What is the velocity of the gases with respect to an observer on the ground?

A river flows with a speed more than the maximum speed with which a person can swim in still water. He intends to cross the river by the shortest possible path, (i.e., he wants to reach the point on the opposite bank which directly opposite to the starting point). Which of the following is correct?

Raindrops are falling vertically with a velocity $10 \mathrm{m} {\mathrm{s}}^{-1}$. To a cyclist moving on a straight road, the raindrops appear to be coming with a velocity of $20 \mathrm{m} {\mathrm{s}}^{-1}$. The velocity of the cyclist is

The distance between two particles is decreasing at a rate of $6 \mathrm{m} {\mathrm{s}}^{-1}$. If these particles travel with same speeds and in the same direction, then the separation increases at the rate of $4 \mathrm{m} {\mathrm{s}}^{-1}$. The particles have speeds as

The speed of a motorboat in still water $20 \mathrm{km} {\mathrm{hr}}^{-1}$ And river flow is $5 \mathrm{km} {\mathrm{hr}}^{-1}$. A float is dropped from the boat when it starts moving upstream. After moving $1.5 \mathrm{km}$ The boat returns. The boat will catch the float (from initial instant) after;

Two trains$\mathrm{A} \mathrm{and} \mathrm{B}$ of length $350\mathrm{m}$ each are moving on two parallel tracks with equal velocities of $100 \mathrm{km}/\mathrm{hr}$ in the same direction, $A$

ahead of $B$. The driver of $B$desires to overtake $\mathrm{A}$.and accelerates by $1 {\mathrm{ms}}^{-2.}$. If after $50 \sec .$ The guard of $B$ just brushes past the driver of $A$, then the original distance between the guard of $\text{B}$ and the driver of $\mathrm{A}$ is :

If a $100 \mathrm{m}$ long train needs $7.2$ seconds to cross an object moving in a direction opposite to the train's direction with a speed of $5 \mathrm{kmph},$ then find the velocity of the train

 Two men, $P$and $Q$ are standing at corners $A$ and $B$ of a square $ABCD$of side $8 \mathrm{m}$. They start moving along the track with constant speed $2 \mathrm{m} {\mathrm{s}}^{-1}$ and $10 \mathrm{m} {\mathrm{s}}^{-1}$, respectively. Find the time when they will meet for the first time.

A man crosses a river perpendicular to the river flow in time $t$ and travels an equal distance down the stream in time $T$. The ratio of man's speed in still water to the speed of river water will be :

A car is standing $\text{200}\text{m}$ 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 ${\text{2 m s}}^{-2}$ and the car has acceleration $4{\text{m s}}^{-2}$ . The car will catch up with the bus after time :

Consider a collection of a large number of particles each moving with speed $v$. The direction of velocity is randomly distributed in the collection. Find the magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection.

A car A is moving with speed $40 \mathrm{km} {\mathrm{h}}^{-1}$ along a straight line $30°$ north of east and another car B is moving with same speed along a straight line $30°$ south of east. The relative velocity of car A as observed from the car B is

A body is thrown up in a lift with a velocity $5\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$ relative to the lift and the time of flight is found to be $0.8\mathrm{ }\mathrm{s}$. The acceleration with which the lift is moving up is $(g=10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-2})$