Motion in a Plane - Comprehensive Study Guide

Introduction to Motion in Two and Three Dimensions

• 1D vs. Multi-dimensional Kinematics:

  • In one-dimensional motion (along a straight line), directional aspects are fully specified using positive ($+$ ) and negative ($-$ ) signs because only two directions are possible.

  • In two-dimensional motion (in a plane) or three-dimensional motion (in space), specifying direction requires vectors.

• Scope of Two-Dimensional Kinematics:

  • Vector algebra provides the foundation for defining velocity and acceleration in two dimensions.

  • Two-dimensional motion with constant acceleration includes important applications such as projectile motion and uniform circular motion.

  • Kinematic equations developed for two-dimensional motion can be directly extended to three-dimensional motion.

Scalars and Vectors

• Scalars:

  • Defined as physical quantities that possess magnitude only and are specified completely by a single real number along with a proper unit.

  • Scalar quantities follow the ordinary rules of algebra (addition, subtraction, multiplication, and division).

  • Addition and subtraction of scalar quantities are meaningful only if the quantities have identical units; multiplication and division can be performed on scalars with different units.

  • Examples of Scalars:

  • Distance between two points.

  • Mass of an object.

  • Temperature of a body (e.g., maximum temperature of $35.6\,^{\circ}\text{C}$  and minimum temperature of $24.2\,^{\circ}\text{C}$  yield a scalar difference of $11.4\,^{\circ}\text{C}$ ).

  • Time at which an event occurs.

  • Perimeter of a rectangle (e.g., sides of $1.0\,\text{m}$  and $0.5\,\text{m}$  yield a scalar perimeter $1.0\,\text{m} + 0.5\,\text{m} + 1.0\,\text{m} + 0.5\,\text{m} = 3.0\,\text{m}$ ).

  • Volume and density (e.g., a uniform aluminum cube of side $10\,\text{cm}$  has volume $10^{-3}\,\text{m}^3$  and mass $2.7\,\text{kg}$ , yielding a scalar density of $2.7 \times 10^3\,\text{kg}\,\text{m}^{-3}$ ).

• Vectors:

  • Defined as physical quantities that possess both a magnitude and a direction, and satisfy the triangle law of addition or equivalently the parallelogram law of addition.

  • Examples of Vectors: Displacement, velocity, acceleration, and force.

  • Vector Notation:

  • Printed in boldface type (e.g., $\mathbf{v}$ , $\mathbf{A}$ , $\mathbf{a}$ ).

  • Handwritten with an arrow placed above the letter (e.g., $\vec{v}$ , $\vec{A}$ ).

  • Magnitude (absolute value) is denoted by lightface type or vertical bars: $|\mathbf{v}| = v$  or $|\mathbf{A}| = A$ .

• Position and Displacement Vectors:

  • Position Vector: Drawn from a chosen origin $\mathbf{O}$  to the particle's position $P$  at time $t$ , denoted as $\mathbf{r} = \mathbf{OP}$ . The length of the line segment represents magnitude, and the arrow specifies direction.

  • Displacement Vector: When a particle moves from position $P$  (at time $t$ ) to position $P'$  (at time $t'$ ), the displacement vector $\mathbf{PP}'$  (or $\Delta\mathbf{r}$ ) is the straight line drawn from initial position $P$  (tail) to final position $P'$  (tip).

  • Displacement is independent of the actual path taken ($PABCQ$ , $PDQ$ , or $PBEFQ$ ); it depends solely on the initial and final position coordinates.

  • The magnitude of displacement is always less than or equal to the actual path length traversed by the object.

• Equality of Vectors:

  • Two vectors $\mathbf{A}$  and $\mathbf{B}$  are defined as equal ($\mathbf{A} = \mathbf{B}$ ) if and only if they have both equal magnitude and identical direction.

  • Free Vectors: Vectors that can be translated parallel to themselves without changing their physical meaning.

  • Localised Vectors: Vectors whose line of application or point of application is fixed in a physical context.

Multiplication of Vectors by Real Numbers

• Multiplication by a Positive Real Number:

  • Multiplying a vector $\mathbf{A}$  by a positive scalar $\lambda > 0$  results in a vector $\lambda \mathbf{A}$  whose direction is identical to $\mathbf{A}$  and whose magnitude is $|\lambda \mathbf{A}| = \lambda |\mathbf{A}|$ .

  • Example: Multiplying $\mathbf{A}$  by $2$  yields $2\mathbf{A}$  with double the magnitude in the same direction.

• Multiplication by a Negative Real Number:

  • Multiplying a vector $\mathbf{A}$  by a negative scalar $-\lambda$  ($\lambda > 0$ ) produces a vector directed opposite to $\mathbf{A}$  with a magnitude of $\lambda |\mathbf{A}|$ .

  • Example: Multiplying $\mathbf{A}$  by $-1$  produces a vector of equal magnitude pointing in the opposite direction ($-\mathbf{A}$ ); multiplying by $-1.5$  produces $-1.5\mathbf{A}$  with $1.5$  times the magnitude in the opposite direction.

• Dimensional Changes:

  • If $\lambda$  is a scalar physical quantity possessing its own dimensions, the dimension of $\lambda \mathbf{A}$  is the product of the dimensions of $\lambda$  and $\mathbf{A}$ .

  • Example: Multiplying a velocity vector by a time duration scalar yields a displacement vector.

Addition and Subtraction of Vectors — Graphical Method

• Triangle Method (Head-to-Tail Method):

  • To add vector $\mathbf{B}$  to vector $\mathbf{A}$ , place the tail of $\mathbf{B}$  at the head of $\mathbf{A}$ .

  • The resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$  is drawn from the tail of $\mathbf{A}$  to the head of $\mathbf{B}$ , forming the third side of a triangle.

• Properties of Vector Addition:

  • Commutative Law: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$

  • Associative Law: $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$

• Null Vector (Zero Vector):

  • Adding two equal and opposite vectors yields a null vector: $\mathbf{A} + (-\mathbf{A}) = \mathbf{0}$ .

  • Magnitude: $|\mathbf{0}| = 0$ .

  • Direction: Indeterminate / non-specifiable.

  • Algebraic Properties:     $\mathbf{A} + \mathbf{0} = \mathbf{A}$

    $\lambda \mathbf{0} = \mathbf{0}$

    $0 \mathbf{A} = \mathbf{0}$

• Physical Meaning: Represents zero change, such as the net displacement of an object moving along a closed loop back to its starting point.

  • Vector Subtraction:

• Subtraction of vector $\mathbf{B}$  from $\mathbf{A}$  is defined as the addition of $\mathbf{A}$  and -$ \mathbf{B} :     $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$

  • Parallelogram Method:

• Bring the tails of vectors $\mathbf{A}$  and $\mathbf{B}$  to a common origin $\mathbf{O}$ .

• Complete a parallelogram using lines parallel to $\mathbf{A}$  and $\mathbf{B}$  passing through their respective heads.

• The resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$  is represented by the diagonal passing through the common origin $\mathbf{O}$ .

Resolution of Vectors and Unit Vectors

• General Vector Resolution:

  • Any vector $\mathbf{A}$  lying in a plane can be uniquely resolved into two component vectors along non-zero, non-collinear vectors $\mathbf{a}$  and $\mathbf{b}$  in the same plane:     $\mathbf{A} = \lambda \mathbf{a} + \mu \mathbf{b}$

    where $\lambda$  and $\mu$  are real numbers.

• Unit Vectors:

  • Defined as a dimensionless, unitless vector of magnitude $1$  pointing in a specified direction.

  • Used strictly to specify direction.

  • Standard Cartesian unit vectors along $x$ , $y$ , and $z$  axes are written as $\hat{\mathbf{i}}$ , $\hat{\mathbf{j}}$ , and $\hat{\mathbf{k}}$  respectively:     $|\hat{\mathbf{i}}| = |\hat{\mathbf{j}}| = |\hat{\mathbf{k}}| = 1$

  • Expressing a general vector $\mathbf{A}$  using its unit direction vector $\hat{\mathbf{n}}$ :     $\mathbf{A} = |\mathbf{A}| \hat{\mathbf{n}}$

• Rectangular Components in Two Dimensions:

  • A vector $\mathbf{A}$  in the $x\text{-}y$  plane resolved along orthogonal unit vectors $\hat{\mathbf{i}}$  and $\hat{\mathbf{j}}$ :     $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$

  • Component magnitudes in terms of magnitude $A$  and inclination angle $\theta$  relative to the $x$ -axis:     $A_x = A \cos(\theta)$

    $A_y = A \sin(\theta)$

• Reconstructing magnitude $A$  and direction $\theta$  from components $A_x$  and $A_y$ :     $A = \sqrt{A_x^2 + A_y^2}$

    $\tan(\theta) = \frac{A_y}{A_x} \implies \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Rectangular Components in Three Dimensions:

  • Direction cosines relative to spatial angles $\alpha$  (with $x$ -axis), $\beta$  (with $y$ -axis), and $\gamma$  (with $z$ -axis):     $A_x = A \cos(\alpha)$

    $A_y = A \cos(\beta)$

    $A_z = A \cos(\gamma)$

• Vector representation in 3D:     $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}} + A_z \hat{\mathbf{k}}$

• Magnitude in 3D:     $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$

• Position vector in 3D:     $\mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}$

Vector Addition — Analytical Method

• Addition via Components (2D):

  • For vectors $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$  and $\mathbf{B} = B_x \hat{\mathbf{i}} + B_y \hat{\mathbf{j}}$ :     $\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x)\hat{\mathbf{i}} + (A_y + B_y)\hat{\mathbf{j}}$

    $R_x = A_x + B_x$

    $R_y = A_y + B_y$

• Addition via Components (3D):

  • For $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}} + A_z \hat{\mathbf{k}}$  and $\mathbf{B} = B_x \hat{\mathbf{i}} + B_y \hat{\mathbf{j}} + B_z \hat{\mathbf{k}}$ :     $R_x = A_x + B_x$

    $R_y = A_y + B_y$

    $R_z = A_z + B_z$

• Linear Combinations:

  • For $\mathbf{T} = \mathbf{a} + \mathbf{b} - \mathbf{c}$ :     $T_x = a_x + b_x - c_x$

    $T_y = a_y + b_y - c_y$

    $T_z = a_z + b_z - c_z$

• Law of Cosines and Law of Sines:

  • For two vectors $\mathbf{A}$  and $\mathbf{B}$  with an enclosed angle $\theta$ :

  • Law of Cosines (Magnitude of Resultant $\mathbf{R}$ ):       $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$

  • Law of Sines:       $\frac{R}{\sin(\theta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)}$

      where $\alpha$  is the angle made by $\mathbf{R}$  with vector $\mathbf{A}$ , and $\beta$  is the angle made by $\mathbf{R}$  with vector $\mathbf{B}$ . - Direction angle $\alpha$ :       $\tan(\alpha) = \frac{B \sin(\theta)}{A + B \cos(\theta)}$

Motion in a Plane: Position, Displacement, Velocity, and Acceleration

• Position Vector and Displacement:

  • Position vector at coordinates $(x, y)$ :     $\mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}}$

  • Displacement between $\mathbf{r}$  at time $t$  and $\mathbf{r}'$  at time $t'$ :     $\Delta\mathbf{r} = \mathbf{r}' - \mathbf{r} = (x' - x)\hat{\mathbf{i}} + (y' - y)\hat{\mathbf{j}} = \Delta x \hat{\mathbf{i}} + \Delta y \hat{\mathbf{j}}$

• Average and Instantaneous Velocity:

  • Average Velocity:     $\bar{\mathbf{v}} = \frac{\Delta\mathbf{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{\mathbf{i}} + \frac{\Delta y}{\Delta t}\hat{\mathbf{j}} = \bar{v}_x \hat{\mathbf{i}} + \bar{v}_y \hat{\mathbf{j}}$

    The direction of $\bar{\mathbf{v}}$  is identical to the direction of $\Delta\mathbf{r}$ .

• Instantaneous Velocity:     $\mathbf{v} = \lim_{\Delta t \rightarrow 0} \frac{\Delta\mathbf{r}}{\Delta t} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \frac{\mathrm{d}x}{\mathrm{d}t}\hat{\mathbf{i}} + \frac{\mathrm{d}y}{\mathrm{d}t}\hat{\mathbf{j}} = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}}$

    The instantaneous velocity vector $\mathbf{v}$  at any point along a path is always directed tangentially to the path at that point in the direction of motion.

• Magnitude of Instantaneous Velocity (Speed):     $v = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$

• Direction angle $\theta$  relative to $x$ -axis:     $\tan(\theta) = \frac{v_y}{v_x} \implies \theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

  • Average and Instantaneous Acceleration:

• Average Acceleration:     $\bar{\mathbf{a}} = \frac{\Delta\mathbf{v}}{\Delta t} = \frac{\Delta v_x}{\Delta t}\hat{\mathbf{i}} + \frac{\Delta v_y}{\Delta t}\hat{\mathbf{j}} = \bar{a}_x \hat{\mathbf{i}} + \bar{a}_y \hat{\mathbf{j}}$

• Instantaneous Acceleration:     $\mathbf{a} = \lim_{\Delta t \rightarrow 0} \frac{\Delta\mathbf{v}}{\Delta t} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = a_x \hat{\mathbf{i}} + a_y \hat{\mathbf{j}}$

    where:     $a_x = \frac{\mathrm{d}v_x}{\mathrm{d}t} = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}$

    $a_y = \frac{\mathrm{d}v_y}{\mathrm{d}t} = \frac{\mathrm{d}^2 y}{\mathrm{d}t^2}$

• In two or three dimensions, velocity and acceleration vectors do not need to be collinear; they can form any angle between $0^\circ$  and $180^\circ$ .

Motion in a Plane with Constant Acceleration

• Kinematic Equations for Constant Acceleration:

  • Velocity as a function of time:     $\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$

    In component form:     $v_x = v_{0x} + a_x t$

    $v_y = v_{0y} + a_y t$

• Position vector as a function of time:     $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2$

    In component form:     $x = x_0 + v_{0x}t + \frac{1}{2}a_x t^2$

    $y = y_0 + v_{0y}t + \frac{1}{2}a_y t^2$

• Independence of Perpendicular Motions:

  • Motion in two dimensions under constant acceleration is mathematically equivalent to two independent, simultaneous one-dimensional motions occurring along mutually perpendicular axes.

Projectile Motion

• Definition and Principles:

  • A projectile is any object launched into space that continues in motion under the influence of gravity alone (neglecting air resistance).

  • Historical Principle (Galileo, 1632): Projectile motion consists of two independent components:

  1. A horizontal component with constant velocity ($a_x = 0$ ).

  2. A vertical component with constant downward acceleration due to gravity ($a_y = -g$ ).

• Initial Conditions:

  • Initial launch speed $v_0$  at elevation angle $\theta_0$  relative to horizontal:     $v_{0x} = v_0 \cos(\theta_0)$

    $v_{0y} = v_0 \sin(\theta_0)$

• Initial coordinates at origin: $x_0 = 0$ , $y_0 = 0$ .

  • Kinematic Equations at Time $t$:

• Horizontal position: $x = (v_0 \cos(\theta_0)) t$

• Vertical position: $y = (v_0 \sin(\theta_0)) t - \frac{1}{2}g t^2$

• Horizontal velocity component: $v_x = v_{0x} = v_0 \cos(\theta_0)$  (constant throughout flight).

• Vertical velocity component: $v_y = v_0 \sin(\theta_0) - g t$ .

  • Equation of Trajectory (Path Shape):

• Substituting $t = \frac{x}{v_0 \cos(\theta_0)}$  into the vertical position equation yields:     $y = \tan(\theta_0) x - \frac{g}{2 v_0^2 \cos^2(\theta_0)} x^2$

• This equation is of the form $y = ax + bx^2$  (where $a$  and $b$  are constants), demonstrating that the trajectory of a projectile is a parabola.

• Key Projectile Formulas:

  • Time to Maximum Height ($t_m$):     At peak height, $v_y = 0$ :     $t_m = \frac{v_0 \sin(\theta_0)}{g}$

  • Total Time of Flight ($T_f$):     Setting $y = 0$  at impact:     $T_f = \frac{2 v_0 \sin(\theta_0)}{g} = 2 t_m$

  • Maximum Height Reached ($h_m$):     Substituting $t = t_m$  into $y(t)$ :     $h_m = \frac{v_0^2 \sin^2(\theta_0)}{2g}$

  • Horizontal Range ($R$):     Total horizontal distance covered during total time of flight $T_f$ :     $R = v_{0x} T_f = (v_0 \cos(\theta_0)) \left(\frac{2 v_0 \sin(\theta_0)}{g}\right) = \frac{v_0^2 \sin(2\theta_0)}{g}$

  • Maximum Horizontal Range ($R_{\max}$):     For a given launch speed $v_0$ , $R$  is maximized when $\sin(2\theta_0) = 1 \implies \theta_0 = 45^\circ$ :     $R_{\max} = \frac{v_0^2}{g}$

  • Symmetry of Ranges:     Complementary elevation angles $\theta_0 = 45^\circ + \alpha$  and $\theta_0 = 45^\circ - \alpha$  yield identical horizontal ranges because $\sin(90^\circ + 2\alpha) = \sin(90^\circ - 2\alpha) = \cos(2\alpha)$ .

Uniform Circular Motion

• Definition:

  • Motion of an object traversing a circular path of radius $R$  at a constant speed $v$ .

  • Although speed is constant, velocity continuously changes direction (tangential to the circular arc at every point), causing acceleration.

• Centripetal Acceleration ($a_c$):

  • Direction: Always points along the radius inward toward the center of the circle ("center-seeking").

  • Magnitude:     $a_c = \frac{v^2}{R}$

  • Published by Christiaan Huygens in 1673.

  • Note: Centripetal acceleration vector is not constant because its direction continuously changes as the particle moves, even though its magnitude is constant.

• Angular Speed ($\omega$):

  • Rate of change of angular displacement $\Delta\theta$ :     $\omega = \frac{\Delta\theta}{\Delta t}$

  • Relation to linear speed $v$  (with arc length $\Delta s = R \Delta\theta$ ):     $v = \omega R$

  • Centripetal acceleration in terms of angular speed:     $a_c = \omega^2 R$

• Period ($T$) and Frequency ($\nu$):

  • Period ($T$ ): Time required to complete one full revolution.

  • Frequency ($\nu = 1/T$ ): Number of revolutions per unit time.

  • Linear speed in terms of period and frequency:     $v = \frac{2\pi R}{T} = 2\pi R \nu$

  • Angular speed in terms of frequency:     $\omega = 2\pi \nu = \frac{2\pi}{T}$

  • Centripetal acceleration in terms of frequency:     $a_c = 4\pi^2 \nu^2 R$

Summary and Key Formulae

• Scalar vs Vector:

  • Scalars: Magnitude only, combined by standard arithmetic.

  • Vectors: Magnitude and direction, combined by vector algebra.

• Vector Addition & Resolution:

  • Commutative: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$

  • Associative: $(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})$

  • 2D Resolution: $\mathbf{A} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}}$  with $A_x = A \cos(\theta)$ , $A_y = A \sin(\theta)$ , $A = \sqrt{A_x^2 + A_y^2}$ , $\tan(\theta) = A_y / A_x$ .

  • Resultant Magnitude (Law of Cosines): $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$

  • Law of Sines: $\frac{R}{\sin(\theta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)}$

• 2D Kinematics:

  • Position: $\mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}}$

  • Instantaneous Velocity: $\mathbf{v} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}}$

  • Instantaneous Acceleration: $\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = a_x \hat{\mathbf{i}} + a_y \hat{\mathbf{j}}$

  • Constant Acceleration Equations:     $\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$

    $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2$

• Projectile Motion Equations:

  • Horizontal Position: $x = (v_0 \cos(\theta_0)) t$

  • Vertical Position: $y = (v_0 \sin(\theta_0)) t - \frac{1}{2}g t^2$

  • Trajectory Equation: $y = \tan(\theta_0)x - \frac{g}{2v_0^2 \cos^2(\theta_0)}x^2$

  • Time of Peak Height: $t_m = \frac{v_0 \sin(\theta_0)}{g}$

  • Time of Flight: $T_f = \frac{2 v_0 \sin(\theta_0)}{g}$

  • Peak Height: $h_m = \frac{v_0^2 \sin^2(\theta_0)}{2g}$

  • Horizontal Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$

• Uniform Circular Motion Equations:

  • Centripetal Acceleration: $a_c = \frac{v^2}{R} = \omega^2 R = 4\pi^2 \nu^2 R$

  • Speed & Angular Speed Relation: $v = \omega R = 2\pi \nu R$

Points to Ponder

• Path length traversed between two points is generally greater than or equal to the magnitude of displacement. They are equal if and only if the path is a straight line without direction reversal.

• Average speed is greater than or equal to the magnitude of average velocity over a given time interval.

• Kinematic equations for constant acceleration do not apply to uniform circular motion because the direction of acceleration changes continuously.

• Resultant velocity of an object subject to two velocities $\mathbf{v}_1$  and $\mathbf{v}_2$  is $\mathbf{v} = \mathbf{v}_1 + \mathbf{v}_2$ . Relative velocity of object 1 with respect to object 2 is $\mathbf{v}_{12} = \mathbf{v}_1 - \mathbf{v}_2$ .

• The net acceleration in circular motion is directed strictly toward the center only if the speed is constant (uniform circular motion).

• Trajectory shape is determined by both acceleration and initial conditions (initial position and initial velocity).

Solved Examples

• Example 3.1:

  • Problem: Rain is falling vertically at $35\,\text{m s}^{-1}$ . Wind starts blowing east to west at $12\,\text{m s}^{-1}$ . Determine the direction an umbrella should be held.

  • Solution:

  • Rain velocity $\mathbf{v}_r$  (downward), wind velocity $\mathbf{v}_w$  (westward).

  • Resultant speed:       $R = \sqrt{v_r^2 + v_w^2} = \sqrt{35^2 + 12^2} = \sqrt{1225 + 144} = \sqrt{1369} = 37\,\text{m s}^{-1}$

  • Angle $\theta$  made by resultant $\mathbf{R}$  with vertical:       $\tan(\theta) = \frac{v_w}{v_r} = \frac{12}{35} \approx 0.343 \implies \theta = \tan^{-1}(0.343) \approx 19^\circ$

  • Umbrella should be held in the vertical plane at an angle of about $19^\circ$  with the vertical toward the east.

• Example 3.2:

  • Problem: Find magnitude and direction of resultant of two vectors $\mathbf{A}$  and $\mathbf{B}$  at angle $\theta$ .

  • Solution:

  • Magnitude: $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$  (Law of Cosines).

  • Direction angle $\alpha$  relative to $\mathbf{A}$ :       $\tan(\alpha) = \frac{B \sin(\theta)}{A + B \cos(\theta)} \implies \alpha = \tan^{-1}\left(\frac{B \sin(\theta)}{A + B \cos(\theta)}\right)$

• Example 3.3:

  • Problem: Motorboat racing north at $25\,\text{km/h}$ , water current at $10\,\text{km/h}$  in direction $60^\circ$  east of south. Find resultant velocity.

  • Solution:

  • Angle between boat velocity $\mathbf{v}_b$  (north) and current velocity $\mathbf{v}_c$  ($60^\circ$  east of south): $\theta = 180^\circ - 60^\circ = 120^\circ$ .

  • Resultant magnitude using Law of Cosines:       $R = \sqrt{v_b^2 + v_c^2 + 2 v_b v_c \cos(120^\circ)} = \sqrt{25^2 + 10^2 + 2(25)(10)(-0.5)} = \sqrt{625 + 100 - 250} = \sqrt{475} \approx 22\,\text{km/h}$

  • Angle $\phi$  relative to north using Law of Sines:       $\frac{R}{\sin(120^\circ)} = \frac{v_c}{\sin(\phi)} \implies \sin(\phi) = \frac{v_c \sin(120^\circ)}{R} = \frac{10 \times 0.866}{21.8} \approx 0.397 \implies \phi \approx 23.4^\circ$  east of north.

• Example 3.4:

  • Problem: Position $\mathbf{r}(t) = 3.0t \hat{\mathbf{i}} + 2.0t^2 \hat{\mathbf{j}} + 5.0 \hat{\mathbf{k}}$ .

  • (a) Find $\mathbf{v}(t)$  and $\mathbf{a}(t)$ .

  • (b) Find magnitude and direction of $\mathbf{v}(t)$  at $t = 1.0\,\text{s}$ .

  • Solution:

  • (a) $\mathbf{v}(t) = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = 3.0 \hat{\mathbf{i}} + 4.0t \hat{\mathbf{j}}$ .       $\mathbf{a}(t) = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = 4.0 \hat{\mathbf{j}}\,\text{m s}^{-2}$ .

  • (b) At $t = 1.0\,\text{s}$ : $\mathbf{v} = 3.0 \hat{\mathbf{i}} + 4.0 \hat{\mathbf{j}}$ .       Magnitude: $v = \sqrt{3.0^2 + 4.0^2} = 5.0\,\text{m s}^{-1}$ .       Direction: $\theta = \tan^{-1}\left(\frac{4.0}{3.0}\right) \approx 53^\circ$  with $x$-axis.

• Example 3.5:

  • Problem: Particle starts at origin at $t = 0$  with $\mathbf{v}_0 = 5.0 \hat{\mathbf{i}}\,\text{m/s}$  and constant acceleration $\mathbf{a} = (3.0 \hat{\mathbf{i}} + 2.0 \hat{\mathbf{j}})\,\text{m/s}^2$ .

  • (a) Find $y$-coordinate when $x = 84\,\text{m}$ .

  • (b) Find speed at this time.

  • Solution:

  • (a) Position $\mathbf{r}(t) = \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2 = (5.0t + 1.5t^2)\hat{\mathbf{i}} + 1.0t^2 \hat{\mathbf{j}}$ .       $x(t) = 5.0t + 1.5t^2 = 84 \implies 1.5t^2 + 5.0t - 84 = 0 \implies 3t^2 + 10t - 168 = 0$ .       Solving quadratic: $(3t + 28)(t - 6) = 0 \implies t = 6\,\text{s}$ .       At $t = 6\,\text{s}$ : $y = 1.0(6)^2 = 36.0\,\text{m}$ .

  • (b) Velocity at $t = 6\,\text{s}$ :       $\mathbf{v}(t) = (5.0 + 3.0t)\hat{\mathbf{i}} + 2.0t \hat{\mathbf{j}}$

      At $t = 6\,\text{s}$ : $\mathbf{v} = 23.0 \hat{\mathbf{i}} + 12.0 \hat{\mathbf{j}}$ .       Speed $v = \sqrt{23^2 + 12^2} = \sqrt{529 + 144} = \sqrt{673} \approx 26\,\text{m s}^{-1}$ .

• Example 3.6:

  • Problem: Prove Galileo's statement that ranges are equal for elevation angles exceeding or falling short of $45^\circ$  by equal amounts $\alpha$ .

  • Solution:

  • Launch angles: $\theta_0 = 45^\circ + \alpha$  and $\theta_0 = 45^\circ - \alpha$ .

  • Double angles $2\theta_0$ : $90^\circ + 2\alpha$  and $90^\circ - 2\alpha$ .

  • Since $\sin(90^\circ + 2\alpha) = \cos(2\alpha)$  and $\sin(90^\circ - 2\alpha) = \cos(2\alpha)$ , the ranges are equal.

• Example 3.7:

  • Problem: Stone thrown horizontally at $15\,\text{m s}^{-1}$  from cliff height $490\,\text{m}$ . Find time to hit ground and impact speed ($g = 9.8\,\text{m s}^{-2}$ ).

  • Solution:

  • Vertical motion: $y(t) = -\frac{1}{2}g t^2 \implies -490 = -4.9 t^2 \implies t^2 = 100 \implies t = 10\,\text{s}$ .

  • Velocity components at impact:       $v_x = v_{0x} = 15\,\text{m s}^{-1}$

      $v_y = -gt = -9.8 \times 10 = -98\,\text{m s}^{-1}$

- Impact speed: $v = \sqrt{v_x^2 + v_y^2} = \sqrt{15^2 + (-98)^2} = \sqrt{225 + 9604} = \sqrt{9829} \approx 99\,\text{m s}^{-1}$

.

• Example 3.8:

  • Problem: Ball thrown at $28\,\text{m s}^{-1}$  at $30^\circ$  above horizontal. Calculate maximum height, time of flight, and horizontal range.

  • Solution:

  • (a) Maximum height: $h_m = \frac{v_0^2 \sin^2(\theta_0)}{2g} = \frac{28^2 \sin^2(30^\circ)}{2 \times 9.8} = \frac{784 \times 0.25}{19.6} = 10.0\,\text{m}$ .

  • (b) Time of flight: $T_f = \frac{2 v_0 \sin(\theta_0)}{g} = \frac{2 \times 28 \times 0.5}{9.8} = \frac{28}{9.8} \approx 2.9\,\text{s}$ .

  • (c) Horizontal range: $R = \frac{v_0^2 \sin(2\theta_0)}{g} = \frac{28^2 \sin(60^\circ)}{9.8} = \frac{784 \times 0.866}{9.8} \approx 69\,\text{m}$ .

• Example 3.9:

  • Problem: Insect in circular groove of radius $R = 12\,\text{cm}$  completes 7 revolutions in $100\,\text{s}$ . Find angular speed, linear speed, and acceleration.

  • Solution:

  • (a) Angular speed: $\omega = \frac{2\pi \nu} = \frac{2\pi \times 7}{100} \approx 0.44\,\text{rad/s}$ .       Linear speed: $v = \omega R = 0.44 \times 12 = 5.3\,\text{cm s}^{-1}$ .

  • (b) Acceleration vector is not constant because its direction changes continuously (inward toward center).       Magnitude of centripetal acceleration: $a_c = \omega^2 R = (0.44)^2 \times 12 \approx 2.3\,\text{cm s}^{-2}$ .

Conceptual Questions and Exercises

• Classification of Quantities (Scalar vs Vector):

  • Scalars: Volume, mass, speed, density, number of moles, angular frequency, work, current, temperature, pressure, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

  • Vectors: Acceleration, velocity, displacement, angular velocity, force, angular momentum, linear momentum, electric field, magnetic moment, relative velocity, impulse.

• Vector Inequalities:

  • $|\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|$  (Equality holds when $\mathbf{a}$  and $\mathbf{b}$  act along the same direction).

  • $|\mathbf{a} + \mathbf{b}| \ge ||\mathbf{a}| - |\mathbf{b}||$  (Equality holds when $\mathbf{a}$  and $\mathbf{b}$  act along opposite directions).

  • $|\mathbf{a} - \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|$  (Equality holds when $\mathbf{a}$  and $\mathbf{b}$  act along opposite directions).

  • $|\mathbf{a} - \mathbf{b}| \ge ||\mathbf{a}| - |\mathbf{b}||$  (Equality holds when $\mathbf{a}$  and $\mathbf{b}$  act along the same direction).