Motion In A Plane Class 11 Physics Chapter Notes | YoLearn.ai

Vectors: The Language of Motion in 2D

Understanding vectors is paramount when dealing with motion in a plane. Unlike scalar quantities (which only have magnitude, like mass or time), vectors possess both magnitude and direction. Physical quantities like displacement, velocity, acceleration, and force are vectors.

• Vector Representation: A vector is represented graphically by a directed line segment, where the length represents magnitude and the arrowhead indicates direction. Mathematically, it can be written as $\vec{A}$ or $\mathbf{A}$.
• Types of Vectors:
• Position Vector: Locates a point in space from the origin.
• Displacement Vector: Represents the change in position ($\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$).
• Unit Vector: A vector with magnitude 1, used to specify direction (e.g., $\hat{i}$, $\hat{j}$, $\hat{k}$ along x, y, z axes).
• Null Vector: A vector with zero magnitude and arbitrary direction.
• Vector Addition (Triangle Law & Parallelogram Law):
• If two vectors $\vec{A}$ and $\vec{B}$ are represented by two sides of a triangle taken in order, their resultant $\vec{R}$ is given by the third side taken in opposite order: $\vec{R} = \vec{A} + \vec{B}$.
• If two vectors are represented by the adjacent sides of a parallelogram, their resultant is given by the diagonal passing through their common point.
• Magnitude of resultant: $R = \sqrt{A^2 + B^2 + 2AB \cos\theta}$, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.
• Direction of resultant (angle $\alpha$ with $\vec{A}$): $\tan\alpha = \frac{B \sin\theta}{A + B \cos\theta}$.
• Resolution of Vectors: Any vector can be resolved into two or more components. For a 2D vector $\vec{A}$ making an angle $\theta$ with the x-axis, its rectangular components are $A_x = A \cos\theta$ and $A_y = A \sin\theta$. So, $\vec{A} = A_x \hat{i} + A_y \hat{j}$.
• Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y + A_z B_z$. Result is a scalar. Useful for finding work done or angle between vectors.
• Vector (Cross) Product: $\vec{A} \times \vec{B} = AB \sin\theta \hat{n}$. Result is a vector perpendicular to both $\vec{A}$ and $\vec{B}$, direction given by the right-hand thumb rule. Magnitude is the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$. Useful for finding torque or angular momentum.

Projectile Motion: The Path of a Thrown Object

Uniform Circular Motion (UCM)

Key Formulas and Must-Remember Points

• Vector Addition: $R = \sqrt{A^2 + B^2 + 2AB \cos\theta}$. $\tan\alpha = \frac{B \sin\theta}{A + B \cos\theta}$.
• Scalar Product: $\vec{A} \cdot \vec{B} = AB \cos\theta$. If $\vec{A} \cdot \vec{B} = 0$, vectors are perpendicular.
• Vector Product: $|\vec{A} \times \vec{B}| = AB \sin\theta$. If $\vec{A} \times \vec{B} = 0$, vectors are parallel.
• Projectile Motion Trajectory: $y = x \tan\theta - \frac{g x^2}{2 u^2 \cos^2\theta}$.
• Time of Flight (Projectile): $T = \frac{2 u \sin\theta}{g}$.
• Maximum Height (Projectile): $H = \frac{u^2 \sin^2\theta}{2g}$.
• Horizontal Range (Projectile): $R = \frac{u^2 \sin(2\theta)}{g}$. Max range at $\theta = 45^\circ$.
• Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$. Always directed towards center.
• Centripetal Force: $F_c = \frac{mv^2}{r} = m\omega^2 r$. Always directed towards center.
• Relation between Linear & Angular Velocity: $v = \omega r$.

Essential Definitions for Motion in a Plane

Vector

A physical quantity having both magnitude and direction, obeying vector laws of addition.

Unit Vector

A vector with a magnitude of one, used solely to specify a direction (e.g., $\hat{i}, \hat{j}, \hat{k}$).

Scalar Product (Dot Product)

An operation between two vectors that yields a scalar quantity, defined as $\vec{A} \cdot \vec{B} = AB \cos\theta$.

Vector Product (Cross Product)

An operation between two vectors that yields another vector perpendicular to both, defined as $\vec{A} \times \vec{B} = AB \sin\theta \hat{n}$.

Projectile

Any object that is thrown or projected into the air and is subject only to the force of gravity (ignoring air resistance).

Trajectory

The path followed by a projectile, which is typically parabolic.

Centripetal Acceleration

The acceleration of an object moving in a circular path, directed towards the center of the circle, due to the continuous change in the direction of its velocity.

Angular Velocity

The rate at which an object rotates or revolves relative to another point, measured in radians per second (rad/s).

Worked Example: Projectile Motion

• {"title":"Example: Calculating Projectile Range","bodyMarkdown":"Q: A ball is kicked with an initial velocity of 20 m/s at an angle of $30^\\circ$ with the horizontal. Calculate its horizontal range. (Take $g = 10 \\text{ m/s}^2$).\n\nA: Given: $u = 20 \\text{ m/s}$, $\\theta = 30^\\circ$, $g = 10 \\text{ m/s}^2$.\n\nFormula for Range: $R = \\frac{u^2 \\sin(2\\theta)}{g}$\n\nSubstitute values:\n$R = \\frac{(20)^2 \\sin(2 \\times 30^\\circ)}{10}$\n$R = \\frac{400 \\sin(60^\\circ)}{10}$\n$R = \\frac{400 \\times \\frac{\\sqrt{3}}{2}}{10}$\n$R = \\frac{200 \\sqrt{3}}{10}$\n$R = 20 \\sqrt{3} \\text{ m}$\n\nApproximately, $R \\approx 20 \\times 1.732 = 34.64 \\text{ m}$."}

Exam Tip: Mastering Motion in a Plane

For projectile motion problems, always resolve the initial velocity into horizontal and vertical components first. Remember that horizontal velocity remains constant, while vertical motion is governed by gravity. Pay close attention to the signs for vertical displacement and velocity based on your chosen coordinate system. In vector problems, clearly identify if a dot product or cross product is required; their applications are distinct. For uniform circular motion, remember that velocity changes but speed remains constant, leading to centripetal acceleration. Derivations of projectile motion formulas (Time of Flight, Max Height, Range) are frequently asked, so practice them thoroughly.

Practice Questions with Solutions

• Q1: Can the magnitude of the resultant of two vectors be less than the magnitude of either vector? A1: No, the magnitude of the resultant can never be less than the magnitude of either individual vector. The minimum resultant magnitude occurs when vectors are antiparallel (subtraction), which can be zero, but not less than either vector's magnitude.
• Q2: What is the velocity and acceleration of a projectile at its highest point? A2: At the highest point, the vertical component of velocity is zero, so velocity is purely horizontal ($v_x = u \cos\theta$). The acceleration is always $g$ (acceleration due to gravity) acting vertically downwards.
• Q3: If a particle is moving in a circle with constant speed, is its motion uniform or non-uniform? A3: Its motion is non-uniform in terms of velocity because the direction of velocity continuously changes, implying acceleration. However, it is often referred to as 'uniform circular motion' specifically because its speed is uniform.
• Q4: Under what condition is the horizontal range of a projectile maximum for a given initial speed? A4: The horizontal range is maximum when the projection angle is $45^\circ$ with the horizontal.

Frequently Asked Questions