General Motion in a Plane
By Motion in a plane refers to the movement of an object on a flat surface. The motion can be rectilinear, in which the object moves in a straight line, or curvilinear, in which the object moves in a curved path. It is described in a 2D coordinate system by using vectors (displacement, velocity, acceleration). The projectile motion and circular motion are examples of motion in a plane.
Let a particle moves in a plane. Consider a reference frame with origin O and orthogonal axes $x$ and $y$ in this plane. The location of O and the directions of $x$ and $y$ do not change with time. Let the particle be at a point $\mathrm{P}_1$ at time $t_1$. The position of the particle at time $t_1$ is specified by the position vector \begin{align} \vec{\mathrm{OP}}_1=\vec{r}_1=x_1\,\hat\imath+y_1\,\hat\jmath. \end{align}
Let the particle moves to the point $\mathrm{P}_2$ at time $t_2=t_1+\Delta t$. The position of the particle at time $t_2$ is \begin{align} \vec{\mathrm{OP}}_2=\vec{r}_2=x_2\,\hat\imath+y_2\,\hat\jmath. \end{align} The displacement of the particle in the time interval $t_1$ to $t_2$ is defined as \begin{align} \Delta\vec{r} &=\vec{r_2}-\vec{r_1} \\ &=(x_2-x_1)\hat\imath+(y_2-y_1)\hat\jmath.\nonumber \end{align} The average velocity of the particle in the time interval $t_1$ to $t_2$ is given by \begin{align} \vec{v}_\text{av}&=\frac{\Delta\vec{r}}{\Delta t} \\ &=\frac{x_2-x_1}{\Delta t}\hat\imath+\frac{y_2-y_1}{\Delta t}\hat\jmath \\ &=v_{\text{av,x}}\hat\imath+v_{\text{av,y}}\hat\jmath,\nonumber \end{align} where $v_{\text{av,x}}$ and $v_{\text{av,y}}$ are the average speeds along $x$ and $y$ directions.
The velocity (or instantaneous velocity) of the particle at a time $t$ is defined as \begin{align} \vec{v} &=\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \\ &=\frac{\mathrm{d}x}{\mathrm{d}t}\hat\imath+\frac{\mathrm{d}y}{\mathrm{d}t}\hat\jmath \\ &=v_x\hat\imath+v_y\hat\jmath.\nonumber \end{align}
Let the velocity of the particle changes from $\vec{v}_1$ to $\vec{v}_2$ in the time interval $t_1$ to $t_2$. The average acceleration of the particle in the time interval $t_1$ to $t_2$ is defined as \begin{align} \vec{a}_\text{av}&=\frac{\Delta\vec{v}}{\Delta t} \\ &=\frac{v_{2,x}-v_{1,x}}{\Delta t}\hat\imath+\frac{v_{2,y}-v_{1,y}}{\Delta t}\hat\jmath\nonumber\\ &=a_{\text{av,x}}\hat\imath+a_{\text{av,y}}\hat\jmath,\nonumber \end{align} where $a_{\text{av,x}}$ and $a_{\text{av,y}}$ are the average accelerations along $x$ and $y$ directions.
The acceleration of the particle at time $t$ is defined as \begin{align} \vec{a} &=\frac{\mathrm{d}\vec{v}}{\mathrm{d}t} \\ &=\frac{\mathrm{d}{v_x}}{\mathrm{d}t}\hat\imath+\frac{\mathrm{d}{v_y}}{\mathrm{d}t}\hat\jmath.\nonumber \end{align}
Constant acceleration
Let a particle starts its motion from the origin at time $t=0$ with a velocity $\vec{u}$. If acceleration $\vec{a}$ of the particle is constant then its velocity $\vec{v}$ and position $\vec{r}$ at time $t$ are given by \begin{align} \vec{v}&=\vec{u}+\vec{a}\,t,\nonumber\\ \vec{r}&=\vec{u}\,t+\tfrac{1}{2}\vec{a}\,t^2.\nonumber \end{align} The average velocity of the particle is given by \begin{align} \vec{v}_\text{av}=(\vec{u}+\vec{v})/2. \end{align} The motion in a plane can be treated as two simultaneous rectilinear motions in orthogonal directions.
Problems from IIT JEE
Problem (IIT JEE 2014): Airplanes A and B are flying with constant velocity in the same vertical plane at angles $30^\mathrm{o}$ and $60^\mathrm{o}$ with respect to the horizontal respectively as shown in figure.
Solution: Let $\vec{V}_\text{A}$ and $\vec{V}_\text{B}$ be the velocity vectors of airplane A and B in a frame attached to the ground. The figure shows $\vec{V}_\text{A}$, $\vec{V}_\text{B}$, and $\vec{V}_\text{B/A}$, the velocity of B relative to A.
