Relative Velocity

A boy standing on the foothpath tosses a ball straight up and catches it. The driver of a car passing by moving with uniform velocity sees this.

The trajectory of the ball as seen by the driver will be

A boat crossing a river moves with a velocity $v$ relative to still water. The river is flowing with a velocity $\frac{v}{2}$ with respect to the bank. The angle with respect to the flow direction with which the boat should move to minimise the drift is

An aeroplane is flying in vertical plane at an angle of $30°$ with the horizontal (north) and wind is blowing from west. A package is dropped from an aeroplane. The velocity of the wind if package hits a kite flying in the space with a position vector $\overrightarrow{R}=\left(400\sqrt{3}\hat{\mathrm{i}}+80\hat{\mathrm{j}}+200\hat{\mathrm{k}}\right) \mathrm{m}$ with respect to the point of dropping.

(Here $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$  are the unit vectors along north and vertically up, respectively, and $\hat{\mathrm{k}}$  be the unit vector due east. Assume that the bag is light enough to get carried away by the wind)

An airplane flies between two cities separated by a distance $D$. Assume that the wind blows directly from one city to the other at a speed $V_{\mathrm{A}}$ (as shown) and the speed of the airplane is $V_0$ relative to the air. Find the time taken by the airplane to make a round trip between the two cities (that is to fly from city $A$to city $B$ and then back to city $A$)?

A man can swim certain distance in still water up and down in time $t_1$. If he swims to some distance down stream the river and returns back to same point he takes time $t_2$. Then,

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

A particle $A$ moves along $x$-axis with a uniform velocity of magnitude $10 \mathrm{m} {\mathrm{s}}^{-1}$. Another particle $B$ moves with uniform velocity $20 \mathrm{m} {\mathrm{s}}^{-1}$ along a direction making an angle $60°$ with the positive direction of $x$-axis as shown in the figure. The relative velocity of $B$ with respect to that of $A$ is,

A ship $A$ is moving westwards with a speed of $10 \mathrm{km} {\mathrm{h}}^{-1}$ and a ship $B$ which is $100 \mathrm{km}$ south of $A$ is moving northwards with a speed of $10 \mathrm{km} {\mathrm{h}}^{-1}$. The time after which the distance between them becomes shortest is,

A cuboidal car of height $3 \mathrm{m}$ is slipping on a smooth inclined plane. A bolt released from the roof of car from centre of roof $\left(P\right)$ then distance from centre of roof where bolt hits the floor with respect to car is

For two particles $A$ and $B$, it is given that, ${\overrightarrow{r}}_{\mathrm{A}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}$, ${\overrightarrow{r}}_{\mathrm{B}}=6\hat{\mathrm{i}}+7\hat{\mathrm{j}}$, ${\overrightarrow{v}}_{\mathrm{A}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and ${\overrightarrow{v}}_{\mathrm{B}}=x\hat{\mathrm{i}}-5\hat{\mathrm{j}}$. What is the value of $x$, if, they collide?

A boat, which has a speed of $5 \mathrm{km} {\mathrm{h}}^{-1}$ in still water, crosses a river of width $1 \mathrm{km}$ along the shortest possible path in $15 \mathrm{minutes}$. The velocity of the river water in $\mathrm{km} {\mathrm{h}}^{-1}$ is

Four persons $K\text{,} L\text{,} M\text{,} N$ are standing at the four corners of a square of side $d$. Each person is now moving with uniform speed $v$ in such a way that $K$ always moves towards $L$, then $L$ towards $M$, $M$ towards $N$ and $N$ towards $K$. Then, four persons will meet at a time,

Two cars$A$ and $B$ are moving such that $A$ has a speed of $40 \mathrm{km} {\mathrm{h}}^{-1}$ towards east and $B$ a speed of $40 \mathrm{km} {\mathrm{h}}^{-1}$ towards north. If the initial separation between them is $2 \mathrm{km}$, what is the minimum distance between the cars?

In the climax of a movie, the hero jumps from a helicopter, and the villain, chasing the hero, also jumps from the same level. After some time, when they come at the same horizontal level, the villain fires a bullet horizontally towards the hero. Both were falling with a constant acceleration of $2 \mathrm{m} {\mathrm{s}}^{-2}$, because of a parachute. Assuming the hero to be within the range of the bullet, which of the following is correct?

A man starts running along a straight road with uniform velocity, $u$ and observes that the rain is falling vertically downwards. If he doubles his speed, he finds that the rain is coming at an angle$\mathrm{\theta }$ to the vertical. The velocity of rain, with respect to the ground, is

Rain is falling vertically with a velocity of $3 \mathrm{km} {\mathrm{h}}^{-1}$. A man walks in the rain with a velocity of $4 \mathrm{km} {\mathrm{h}}^{-1}$. The raindrops will fall on the man with a velocity of

 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 swimmer crosses a river, along the line making an angle of $45°$ with the direction of flow. The velocity of the river water is $5 \mathrm{m} {\mathrm{s}}^{-1}$. The swimmer takes $12 \mathrm{seconds}$ to cross the river of width $60 \mathrm{m}$. The velocity of the swimmer with respect to water, will be

A man wearing a hat of extended length $12 \mathrm{cm}$ is running in rain falling vertically downwards with speed $10 \mathrm{m} {\mathrm{s}}^{-1}$. The maximum speed with which the man can run, so that rain drops do not fall on his face (the length of his face below the extended part of the hat is $16 \mathrm{cm}$) will be

A boat is rowed across a river at a rate of $4.5 \mathrm{km} {\mathrm{hr}}^{-1}$. The river flows at the rate of $6 \mathrm{km} {\mathrm{hr}}^{-1}$. The velocity of the boat in $\mathrm{m} {\mathrm{s}}^{-1}$ is