Motion in a Plane

The trajectory of a projectile in a vertical plane is $y=ax-b{x}^2,$ where $a, b$ are constants, and $x$ and $y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. Find the maximum height $H_{\max }$ attained by the projectile and the angle of projection from the horizontal.

A person moves $30\mathrm{m}$ north, then $20\mathrm{m}$ east and finally $30\sqrt{2}\mathrm{m}$ south west. What is the displacement from original position ?

A body is simultaneously given two velocities, one $10 \mathrm{m}{\mathrm{s}}^{-1}$ due east and other $20 \mathrm{m}{\mathrm{s}}^{-1}$due north-west. Calculate the resultant velocity.

A train is moving eastward at $10 \mathrm{m}/\sec$. A waiter is walking eastward at $1.2\mathrm{m}/\sec$ and a fly is charging towards the north across the waiter's tray at $2 \mathrm{m}/\mathrm{s}$. What is the velocity of the fly relative to Earth. 

The position of a particle is given by $\overrightarrow{r}=\left(3t\hat{i}+4t^3\hat{j}+5\hat{k}\right)\mathrm{m}$. Find the magnitude of velocity and acceleration of particle at $t=5s$.

The vector joining the points $A\left(1, 1,-1\right)$ and $B\left(2,-3, 4\right)$ and pointing from $A$ to $B$ is

At which point of a projectile motion, acceleration and velocity are perpendicular to each other?

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

Which of the formula is magnitude of the acceleration vector?

What will be resultant acceleration if $a_x = 4 m/s^2 , a_y= 3 m/s^2?$

Three balls, $A, B$ and $C$ are released and all reach the point $X$ (shown in the figure). Balls $A$ and $B$ are released from two identical structures, one kept on the ground and the other at height $h$ from the ground as shown in the figure. They take time $t_A$ and $t_B$ respectively to reach $X$ (time starts after they leave the end of the horizontal portion of the structure). The ball $C$ is released from a point at height $h$ vertically above $X$ and reaches $X$ in time $t_C$. Choose the correct statement.

A particle starts from the origin with an initial velocity $\overrightarrow{v}=\left(3.0\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ and a constant acceleration $\overrightarrow{a}=\left(-0.5\hat{\mathrm{i}}-1.0\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-2}$. The velocity of the particle when it reaches its maximum y coordinate is

An object is moving in a plane and its trajectory $\overrightarrow{r}\left(t\right)$ (in meter) is given by, $\overrightarrow{r}\left(t\right)=\left(3t^2+2t\right)\hat{\mathrm{i}}+2t\hat{\mathrm{j}}$, where t is time (in second). The angle that the velocity vector makes with the $X$ -axis at $t=0$.

A particle is moving along a circular path of radius $5 \text{m}$ with a uniform speed of $5 {\text{ms}}^{-1}$.What will be the average accelaration when the particle completes half revolution ?

Direction of which of the following vector changes, when a tennis ball falls to the ground and bounces back.

i) Acceleration

ii) Velocity

iii) Displacement

The equations of motion of a projectile are given by $x=36 t\mathrm{ }\mathrm{m}$ and $2 y=96 t-9.8\mathrm{ }{t}^2 \mathrm{m}$. The angle of projection is –

The resultant of two vectors is perpendicular to first vector of magnitude $6 \mathrm{N}$. If the resultant has magnitude $6\sqrt{3}\text{ N}$, then magnitude of second vector is

Aeroplane is coming from north at some height from ground when aeroplane is just above the person; person hears the second coming from north making angle ${60}^o$ with horizontal. Find speed of aeroplane