MOTION IN A PLANE
Frequently Asked Questions
Motion in a plane is motion of a particle in two dimensions, where its position, velocity, and acceleration are represented by vectors in an \(x\text{-}y\) plane.
A scalar quantity is one that has only magnitude and no direction, such as mass, distance, speed, time, or temperature.
A vector quantity has both magnitude and direction, such as displacement, velocity, acceleration, and force.
Position vector \(\vec{r}\) of a particle at \((x,y)\) is given by \(\vec{r}=x\hat{i}+y\hat{j}\) with respect to the origin \(O(0,0)\).
Displacement vector is the change in position: \(\Delta\vec{r}=\vec{r}_2-\vec{r}_1\), independent of the actual path followed.
Average velocity is \(\vec{v}_{\text{avg}}=\frac{\Delta\vec{r}}{\Delta t}\), where \(\Delta\vec{r}\) is displacement in time interval \(\Delta t\).
Instantaneous velocity is \(\vec{v}=\frac{d\vec{r}}{dt}\) and is always tangent to the path at that instant.
Average acceleration is \(\vec{a}_{\text{avg}}=\frac{\Delta\vec{v}}{\Delta t}\), where \(\Delta\vec{v}\) is change in velocity in time \(\Delta t\).
Instantaneous acceleration is \(\vec{a}=\frac{d\vec{v}}{dt}\) and measures the rate of change of velocity vector at a given instant.
If two vectors are represented by two sides of a triangle taken in order, the third side taken in the same order represents their resultant.
If two vectors from the same point form adjacent sides of a parallelogram, the diagonal through that point gives the resultant vector.
For vectors \(\vec{A}\) and \(\vec{B}\) with angle \(\theta\) between them, resultant magnitude is \(R=\sqrt{A^2+B^2+2AB\cos\theta}\).
Vector subtraction \(\vec{A}-\vec{B}\) is defined as \(\vec{A}+(-\vec{B})\), where \(-\vec{B}\) has same magnitude as \(\vec{B}\) but opposite direction.
A unit vector has magnitude 1 and gives only direction; unit vector along \(\vec{A}\) is \(\hat{A}=\frac{\vec{A}}{|\vec{A}|}\)
If \(\vec{A}\) makes angle \(\theta\) with positive \(x\)-axis, then \(A_x=A\cos\theta\), \(A_y=A\sin\theta\), and \(\vec{A}=A_x\hat{i}+A_y\hat{j}\).
Resolution is the process of splitting a vector into mutually perpendicular component vectors whose vector sum equals the original vector.
Two vectors are equal if they have same magnitude and same direction, irrespective of their initial points.
A null vector has zero magnitude and an arbitrary direction, represented by \(\vec{0}\).
With constant acceleration \(\vec{a}\), position is \(\vec{r}=\vec{r}_0+\vec{v}_0 t+\frac{1}{2}\vec{a}t^2\) in vector form.
Velocity is \(\vec{v}=\vec{v}_0+\vec{a}t\) when acceleration \(\vec{a}\) is constant.
By resolving vectors along \(x\) and \(y\) axes so that motion along each axis is treated as independent one-dimensional motion.
Projectile motion is the motion of a body projected into the air, moving under the influence of gravity alone, neglecting air resistance.
Angle of projection \(\theta_0\) is the angle between initial velocity vector \(\vec{v}_0\) and the horizontal direction.
For projection with speed \(v_0\) at angle \(\theta_0\), total time of flight is \(T=\frac{2v_0\sin\theta_0}{g}\).
Maximum height reached is \(H=\frac{v_0^2\sin^2\theta_0}{2g}\).
Horizontal range on level ground is \(R=\frac{v_0^2\sin(2\theta_0)}{g}\).
For a given \(v_0\), range \(R\) is maximum when \(\theta_0=45^\circ\).
The trajectory equation is \(y=x\tan\theta_0-\frac{g x^2}{2v_0^2\cos^2\theta_0}\), representing a parabola.
Two angles \(\theta_1\) and \(\theta_2\) such that \(\theta_1+\theta_2=90^\circ\); for same speed, they give same range.
Air resistance reduces range and maximum height, and makes the descending path steeper than the ascending path.
Uniform circular motion (UCM) is motion in a circular path with constant speed; direction of velocity continuously changes.
Centripetal acceleration is the acceleration directed towards the center of the circular path, with magnitude \(a_c=\frac{v^2}{R}=\omega^2 R\).
For circular motion, linear speed \(v\) and angular speed \(\omega\) are related by \(v=\omega R\).
Velocity at any instant is tangential to the circular path, perpendicular to the radius vector.
Centripetal acceleration always points towards the center of the circular path.
Examples include motion of a satellite in a circular orbit and a stone tied to a string whirled in a horizontal circle.
In uniform circular motion speed is constant and only centripetal acceleration exists; in non-uniform circular motion, speed changes and tangential acceleration also acts.
Relative velocity of B with respect to A is \(\vec{v}_{BA}=\vec{v}_B-\vec{v}_A\).
Boat velocity relative to water and river flow velocity are treated as vectors; resultant gives boat velocity relative to ground.
The swimmer must head upstream such that the component of swimming velocity opposite to river flow cancels the river’s velocity.
Rain velocity and person’s velocity are treated as vectors; resultant rain velocity relative to person gives the direction to hold the umbrella.
Gravity acts vertically; horizontal acceleration is zero, so horizontal velocity remains constant in ideal projectile motion.
Vertical component changes due to constant acceleration \(g\) downward; \(v_y=v_{0y}-gt\).
The trajectory is a parabola because \(y\) depends on \(x^2\) in the equation of motion.
Typical questions include finding optimum angle, range, height, or time of flight for balls, bullets, or stones thrown at an angle.
Numericals often ask for centripetal acceleration, speed, or tension in the string for a mass moving in a horizontal or vertical circle.
Standard problems involve boats crossing rivers with flow or people walking in rain, asking for resultant speed and direction.
Analytical method uses components along coordinate axes: resolve vectors into components, add components algebraically, then recombine to get resultant.
Graphical methods like head-to-tail triangle or parallelogram represent vectors as directed line segments and construct the resultant geometrically.
In two dimensions, both magnitude and direction of physical quantities are important; vectors conveniently handle both and allow component-wise analysis.
Multiplying vector \(\vec{A}\) by scalar \(\lambda\) gives \(\lambda\vec{A}\) with magnitude \(|\lambda|A\); direction is same as \(\vec{A}\) if \(\lambda>0\), opposite if \(\lambda<0\).
Average speed is total path length divided by time; average velocity is displacement divided by time and is a vector.
Centrifuges use high-speed circular motion to create large centripetal acceleration for separating components based on density.
If \(\vec{r}_0=\vec{0}\) and \(\vec{v}_0=v_{0x}\hat{i}+v_{0y}\hat{j}\), then \(\vec{r}(t)=v_{0x}t\,\hat{i}+\left(v_{0y}t-\frac{1}{2}gt^2\right)\hat{j}\).
Principle of independence of motions along perpendicular directions: motion along \(x\) does not affect motion along \(y\) when forces act separately.
The chapter uses vector addition, subtraction, scalar multiplication, and resolution into components; dot and cross products are introduced only qualitatively or in later chapters.
Students often use total velocity instead of components, or forget that \(v_x\) remains constant while \(v_y\) changes with time.
Frequently tested topics are projectile formulas (T, H, R), derivation of centripetal acceleration, vector addition and resolution, and basic relative velocity problems.
Concepts of vectors, projectile motion, and circular motion are foundational for later mechanics, making this chapter crucial for solving advanced kinematics and dynamics problems in JEE/NEET.
