Position Vector, Displacement, Velocity and Acceleration

A football player is moving southward and suddenly turns eastward with the same speed to avoid an opponent. The force that acts on the player while turning is :

 If tangential acceleration vector of a speeding up particle at$P$is $3\hat{\mathrm{i}}+4\stackrel{⏜}{\mathrm{j}} \mathrm{m} {\mathrm{s}}^{-2}$ find the angle $\theta$
 

A particle is moving in a straight line such that its velocity is increasing at $5 {\mathrm{ms}}^{-1}$ per meter. The acceleration of the particle is _____ ${\mathrm{ms}}^{-2}$ at a point where its velocity is $20 {\mathrm{ms}}^{-1}$.

A particle attached to a light wire of length is projected horizontally with a velocity of $\surd 5gl$. Calculate its net acceleration when wire becomes horizontal.
 

Displacement of a particle in periodic motion is expressed as $x\left(t\right)=20 \cos \omega t$. If the time period of particle motion is $4 \mathrm{s}$, then displacement of the particle in $1 \mathrm{s}$ will be

Displacement of a particle is given $\overrightarrow{r}=3t \hat{i}+6 \hat{j}$. Find its velocity vector.

The position vector of a particle is given by $\mathrm{r} = 2\mathrm{t}\hat{\mathrm{i}}-5{\mathrm{t}}^2\hat{\mathrm{j}}$. What is the angle between initial velocity and initial acceleration?

A particle moves in the X-Y plane with a constant acceleration of  $1.5 \mathrm{m}/{\mathrm{s}}^2$ in the direction making an angle of $37°$ with the X-axis. At $t=0$ the particle is at the origin and its velocity is $8 \mathrm{m}/\mathrm{s}$ along the X-axis. Find the velocity and the position of the particle at $t= 4 \mathrm{s}$. 

A particle moves along a straight line such that its velocity $\left(v\right)$ at any time $\left(t\right)$ is given by $v=(2t - 1) \mathrm{m}/\mathrm{s}$, where $t$ is in seconds. Total distance travelled by the particle in first second is

The position of an object changes from $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}) \mathrm{m} \mathrm{to} \overrightarrow{{\mathrm{r}}_1}=\left(4 \hat{\mathrm{i}}+3\widehat{ \mathrm{j}}\right) \mathrm{m}$ in $2 \mathrm{s}$. Find its average velocity.

If the position of the particle is given by $\mathrm{x}=3{\mathrm{t}}^2-5\mathrm{t}+7$ then find the velocity and the acceleration at $\mathrm{t}=2 \mathrm{s}$.

On an open ground, a boy follows a track that turns to his left by an angle of $72°$ after every $10 \mathrm{m}$. Starting from a given turn, the displacement of the boy at the fourth turn will be

A ring of radius $10\mathrm{cm}$ is initially at the coordinate plane such that it touches the positive sides of the $x$ and $y$ axes at $P$ and $Q$ respectively. The ring starts rolling at a speed of $1\mathrm{rev}/\min$ along the positive $x$-axis. The coordinates of the point $Q$ after $2$ minutes and $45$ seconds are

A particle moves in the $X-Y$ plane with a constant acceleration of $1.5 \mathrm{m} {\mathrm{s}}^{-2}$ in the direction making an angle of $37°$ with the $X$-axis. At $t=0$ the particle is at the origin and its velocity is $8.0 \mathrm{m} {\mathrm{s}}^{-1}$ along the $X$-axis. Find the velocity and the position of the particle at $t=4·0 \mathrm{s}$.

A particle is moving on a circular path of radius $r$ with uniform, speed $v$. What is the displacement of the particle after it has described an angle of $60°$?

A particle moves in a straight line so that distance $s=\sqrt{t}$ where $t$ is time, then the acceleration is proportional to

Three particles $A,B$ and $C$ are situated at the vertices of an equilateral triangle. They start moving with equal speeds (remains constant during the whole motion) such that $A$ always heads towards $B,B$ towards $C$ & $C$ towards  $A$. Finally they meet at a point. Find out the angle by which the line connecting $A$ and $B$ rotates with respect to initial orientation in space by the time distance between $A$ and $B$ reduces to half of the initial distance between $A$ and $B$.

In $x-y$ plane a particle is displaced from a point $A\left(1,2,3\right) \mathrm{m}$ to a point $B\left(-1,0,1\right) \mathrm{m}$ in $2\sec$. Find the displacement of the particle and its average velocity in vector form in the given time interval.