Analytical Method of Vector Addition

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

The coordinates of a particle at time $t$ are $x=2t+4t^2$, where $x$ and $y$ are in metre and $t$ is in second. The acceleration of the particle at $t=5 s$ is :

A river is flowing from west to east at speed of $5 \mathrm{m}/\mathrm{minute}$. A man who can swim at rate of $10 \mathrm{m}/\mathrm{minute}$ in still water is standing on south bank of river. He wants to swim across river in the shortest time. He should swim in a direction:

Select the correct alternative. The coordinates of a particle moving in a plane are given by $x=a \cos pt$ and $y=b \sin pt$, where $a, b$ and $p$ are positive constants of appropriate dimensions. Then:

A person aiming to reach just opposite point on bank of stream swims at speed of $0.5 \mathrm{m}/\mathrm{s}$ at angle $120°$ with direction of flow of water. The speed of water in stream is _____.

The velocity of a particle moving on a circular path is $5 cm{s}^{-1}$ towards north at any instant. After traversing one-fourth of the path its velocity is $5 cm{s}^{-1}$ towards east. Indicate the change in velocity in a vector diagram.

A car is going towards north with a velocity $20 m{s}^{-1}$.  After some time it starts to go towards south with velocity $20 m{s}^{-1}$. Determine change in the velocity of the car.

The magnitudes of two mutually perpendicular vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ are $5$ and $7$ respectively. Find the magnitude of $\overrightarrow{P}+\overrightarrow{Q}$ and $\overrightarrow{P}-\overrightarrow{Q}$.

A river flows due north with a velocity of $3 \mathrm{km}/\mathrm{h}$ . A man rows a boat across the river with velocity of $4 \mathrm{km}/\mathrm{h}$ relative to water due east. How long a time is required to cross the river.

A river flows due north with velocity of $3 \mathrm{km}/\mathrm{h}$. A man rows a boat across the river with a velocity of $4 \mathrm{km}/\mathrm{h}$ relative to water due east. If river is $1 \mathrm{km}$ wide, how far north of his starting point will he reach the opposite bank.

A river flows due north with a velocity of $3 \mathrm{km}/\mathrm{h}$. A man rows a boat across river with a velocity of $4 \mathrm{km}/\mathrm{h}$ relative to water due east. What is his velocity relative to the earth?

A $300 \mathrm{m}$ wide river flows with a speed of $3 \mathrm{m}/\mathrm{s}$. A man swims across the river with a velocity of $2 \mathrm{m}/\sec$ directed always perpendicular to the flow of current. How far down the stream (from the starting point) does he reach the opposite bank?

A river flows at $3 \mathrm{m}/\sec$ and $300 \mathrm{m}$ wide A man swims across the river with velocity $2 \mathrm{m}/\sec$ directed always perpendicular to flow of current. In what direction does he actually move relative to the shore.

A  $300 \mathrm{m}$ wide river flows at a speed of $3 \mathrm{m}/\sec$. A man swims across the river with a speed of $2 \mathrm{m}/\sec$ directed always perpendicular to flow of current. How long does it take the man to cross the river?

Show that $\overrightarrow{\mathrm{A}}=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{B}}=12\hat{\mathrm{i}}+9\hat{\mathrm{j}}+3\hat{\mathrm{k}}$ are parallel to each other.

If $\overrightarrow{\mathrm{A}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{B}}=3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ Find out $\left(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\right)\times \left(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}\right)$

The velocity of a body is $100 \mathrm{km}/\mathrm{h}$, $30°$ west of south. Find north and east components by drawing vector diagram.

A force of $500 \mathrm{N}$ is acting towards east and another of $600 \mathrm{N}$ towards north. Subtract the first force from second by drawing a vector diagram.

Two forces of $6 \mathrm{N}$ and $8 \mathrm{N}$ are acting on a point at an angle of $90°$ with each other. Determine the magnitude and direction of the resultant force by drawing a vector diagram.

A particle has an initial velocity $3\hat{i}+4\hat{j}$ and an acceleration of $0.4\hat{i}+0.3\hat{j}$. Its speed after $10 s$ is: