AP Physics 1 Unit 2 Study Guide: Vectors and Two-Dimensional Motion

Fundamental Vector Concepts and Graphical Operations

• Vector Definition and Properties:

  • A vector is a mathematical and physical quantity defined by both a magnitude (size or length) and a direction.

  • The magnitude of a vector is a strictly non-negative scalar quantity ($\text{Magnitude} \ge 0$).

  • Vectors are visually represented as arrows. The length of the arrow is directly proportional to the magnitude of the vector, and the arrowhead specifies its direction.

  • Vector quantities are distinct from scalar quantities, which possess magnitude only (e.g., mass, time, temperature, distance).

  • Proper vector notation uses an arrow above the variable symbol (e.g., $\vec{A}$, $\vec{v}$, $\vec{r}$, $\vec{F}$).

• Vector Equality:

  • Two vectors $\vec{A}$ and $\vec{B}$ are defined as equal ($\vec{A} = \vec{B}$) if and only if they possess identical magnitudes and point in the exact same direction.

  • Vector equality is completely independent of the starting point or location of the vectors in space. Translating a vector arrow without altering its length or orientation preserves its physical identity.

• Displacement Vector vs. Path Traveled:

  • The displacement vector ($\vec{d}$ or $\Delta \vec{r}$) is defined as the straight-line directed distance connecting an initial position to a final position.

  • Displacement depends solely on the initial and final position coordinates and is completely independent of the actual trajectory or distance traveled along the path.

• Vector Addition Principles:

  • The vector sum of two or more vectors is termed the resultant vector ($\vec{R}$).

  • Vector addition is commutative:         $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

  • Vector addition is associative:         $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$

• Graphical Addition Methods:

  • Tip-to-Tail Method:

    • To add vectors graphically, translate the vectors without changing their orientation so that the tail (starting point) of each subsequent vector touches the tip (arrowhead) of the preceding vector.

    • The resultant vector $\vec{R}$ is drawn starting from the tail of the first vector and terminating at the tip of the final added vector.

  • Parallelogram Method:

    • Place the tails of two vectors $\vec{D}$ and $\vec{E}$ at a common origin.

    • Construct a parallelogram using the two vectors as adjacent sides.

    • The resultant vector $\vec{F} = \vec{D} + \vec{E}$ is represented by the diagonal vector extending from the common origin to the opposite corner of the parallelogram.

• Scalar Multiplication and Division:

  • Multiplying a vector $\vec{A}$ by a positive scalar factor $c > 0$ yields a new vector $\vec{B} = c\vec{A}$:

    • The magnitude of $\vec{B}$ is scaled by factor $c$ ($B = cA$).

    • The direction remains unchanged ($\theta_B = \theta_A$).

    • If $c > 1$, the vector arrow is stretched.

    • If $0 < c < 1$, the vector arrow is compressed.

  • Dividing a vector by a scalar $c$ is mathematically equivalent to multiplying by $\frac{1}{c}$.

• Negation and Vector Subtraction:

  • Multiplying a vector $\vec{A}$ by $-1$ produces the negative vector $-\vec{A}$.

  • The vector $-\vec{A}$ possesses the exact same magnitude as $\vec{A}$ ($| -\vec{A} | = | \vec{A} |$) but points in the opposite direction ($180^\circ$ orientation flip).

  • The sum of a vector and its negative produces the zero vector:         $\vec{A} + (-\vec{A}) = \vec{0}$

  • Vector subtraction is performed by adding the negated vector:         $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

  • To perform graphical subtraction, invert the direction of the vector being subtracted (flip its tip and tail) and place it into tip-to-tail configuration with the base vector.

Motion Diagrams and Kinematic Vectors

• Position and Displacement Vectors in Two Dimensions:

  • Position vector $\vec{r}$ defines the location of a particle relative to a chosen coordinate origin in two dimensions.

  • Displacement vector $\Delta \vec{r}$ (or $\vec{d}$) over a time interval $\Delta t = t_f - t_i$ is defined as:         $\Delta \vec{r} = \vec{r}_f - \vec{r}_i$

• Average Velocity Vector:

  • Average velocity $\vec{v}_{avg}$ is defined as the displacement divided by the scalar time interval:         $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \left(\frac{1}{\Delta t}\right) \Delta \vec{r}$

  • Because time interval $\Delta t$ is a positive scalar, the average velocity vector $\vec{v}_{avg}$ always points in the exact same direction as the displacement vector $\Delta \vec{r}$.

• Average Acceleration Vector:

  • Average acceleration $\vec{a}_{avg}$ is defined as the change in velocity divided by the time interval:         $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t}$

  • The acceleration vector $\vec{a}_{avg}$ always points in the exact same direction as the change in velocity vector $\Delta \vec{v} = \vec{v}_f - \vec{v}_i$.

• Interpreting Kinematic Vector Relationships:

  • Speeding Up: Occurs when velocity $\vec{v}$ and acceleration $\vec{a}$ vectors point in the same general direction (angle between them is acute, $\theta < 90^\circ$).

  • Slowing Down: Occurs when velocity $\vec{v}$ and acceleration $\vec{a}$ vectors point in opposing directions (angle between them is obtuse, $90^\circ < \theta \le 180^\circ$).

  • Changing Direction at Constant Speed: Occurs when the acceleration vector $\vec{a}$ is strictly perpendicular to the velocity vector $\vec{v}$ ($\theta = 90^\circ$).

• Uniform Circular Motion:

  • An object moving in a circular path at constant speed is continuously accelerating because the direction of its velocity vector changes continuously.

  • Vector subtraction of consecutive velocity vectors $\Delta \vec{v} = \vec{v}_2 - \vec{v}_1$ demonstrates that the change in velocity vector points inward toward the center of the circle.

  • Consequently, the acceleration vector $\vec{a}$ for uniform circular motion points directly toward the center of the circular trajectory (centripetal acceleration).

Two-Dimensional Coordinate Systems and Component Resolution

• Cartesian Coordinate Frame:

  • A two-dimensional coordinate system consists of two mutually perpendicular axes ($x$ and $y$).

  • Component vectors $\vec{A}_x$ and $\vec{A}_y$ are defined parallel to the $x$-axis and $y$-axis, respectively, such that:         $\vec{A} = \vec{A}_x + \vec{A}_y$

  • Decomposing a vector into its component vectors is the inverse operation of vector addition.

• Scalar Components and Trigonometric Formulas:

  • A vector $\vec{A}$ with magnitude $A$ and angle $\theta$ measured relative to the positive $x$-axis decomposes into scalar components $A_x$ and $A_y$:         $A_x = A \cos(\theta)$         $A_y = A \sin(\theta)$

  • The sign of a scalar component indicates its direction along the corresponding axis:

    • Positive if pointing along the positive coordinate direction.

    • Negative if pointing along the negative coordinate direction.

  • The magnitude of a component vector equals the absolute value of the scalar component:         $|\vec{A}_x| = |A_x$         $|\vec{A}_y| = |A_y|$

  • Trigonometric Rule Caution: Cosine corresponds to the side adjacent to angle $\theta$, and Sine corresponds to the side opposite to angle $\theta$. Cosine is not inherently tied to the $x$-axis, nor Sine to the $y$-axis. If an angle $\phi$ is measured relative to the vertical ($y$-axis):         $C_x = C \sin(\phi)$         $C_y = -C \cos(\phi)$

• Reconstructing Vector Magnitude and Direction:

  • Given scalar components $A_x$ and $A_y$, the overall vector magnitude $A$ is derived using the Pythagorean Theorem:         $A = \sqrt{A_x^2 + A_y^2}$

  • The orientation angle $\theta$ relative to the horizontal is calculated using the inverse tangent function:         $\theta = \tan^{-1}\left(\frac{|A_y|}{|A_x|}\right)$

• Direction Conventions and Cardinal Headings:

  • Standard Cartesian angles are measured counterclockwise from the positive $x$-axis ($+x$).

  • Negative angles represent clockwise rotation from the positive $x$-axis.

  • Cardinal Direction Notation: Specified as "[Angle] [Direction 1] of [Direction 2]". Direction 2 serves as the reference axis from which the angular rotation toward Direction 1 occurs.

    • Example: $30^\circ\text{ South of West}$ means start pointing due West, then rotate $30^\circ$ toward South.

  • Complementary Angle Equivalence: $30^\circ\text{ South of West}$ is identical in orientation to $60^\circ\text{ West of South}$.

• Algebraic Vector Addition via Components:

  • To add arbitrary vectors $\vec{A}, \vec{B}, \vec{C}, \dots$ algebraically:

    1. Decompose each vector into its scalar $x$- and $y$-components using appropriate trigonometry.

    2. Sum all $x$-components to find the resultant $x$-component:             $R_x = A_x + B_x + C_x + \dots$

    3. Sum all $y$-components to find the resultant $y$-component:             $R_y = A_y + B_y + C_y + \dots$

    4. Calculate total resultant magnitude and orientation:             $R = \sqrt{R_x^2 + R_y^2}$             $\theta = \tan^{-1}\left(\frac{|R_y|}{|R_x|}\right)$

Relative Motion and Reference Frame Kinematics

• Reference Frames Defined:

  • A reference frame is a coordinate system used by an observer to measure the position, velocity, and acceleration of physical objects.

  • A reference frame extends infinitely in all space. All observers at rest relative to one another share the exact same reference frame.

  • Reference frame is distinct from point of view; two observers standing on opposite sides of a road watching a car belong to the same reference frame because they are at rest relative to each other.

• Inertial vs. Non-Inertial Reference Frames:

  • Inertial Reference Frame: A frame of reference in which an observer experiences zero acceleration (at rest or moving at a constant velocity vector relative to another inertial frame).

    • Newton's Laws of Motion are strictly valid in all inertial reference frames.

    • Example: An airplane flying at constant altitude and constant velocity is an inertial frame; a ball placed on the cabin floor remains at rest relative to the airplane.

  • Non-Inertial Reference Frame: A reference frame that undergoes acceleration ($\vec{a} \neq 0$).

    • Newton's Laws of Motion break down in non-inertial frames unless fictitious forces are introduced.

    • Example: An airplane accelerating down a runway during takeoff is non-inertial; a ball on the cabin floor rolls backward relative to the plane without any real horizontal contact force applied to it.

  • Earth's surface is a highly accurate approximation of an inertial reference frame for localized laboratory experiments.

• Relative Velocity Subscript Notation:

  • In two-subscript notation $\vec{v}_{AB}$, the first subscript ($A$) represents the object being observed, and the second subscript ($B$) represents the reference frame of the observer.

  • Reversing reference frame subscripts negates the velocity vector:         $\vec{v}_{BA} = -\vec{v}_{AB}$

• Relative Motion Subscript Cancellation Rule:

  • Relative velocities combine linearly by setting up intermediate reference frames:         $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$

  • The inner subscript ($B$) cancels out, leaving the velocity of object $A$ relative to reference frame $C$.

• Galilean Transformation Equations:

  • Consider Frame $S'$ moving with constant velocity $v$ along the $x$-axis relative to Frame $S$.

  • If the origin of $S'$ coincides with $S$ at $t = 0$:         $x' = x - vt$         $u' = u - v$         $u = u' + v$

  • where $u$ is the velocity of an object measured in Frame $S$, $u'$ is the velocity measured in Frame $S'$, and $v$ is the relative velocity between frames $S$ and $S'$.

Two-Dimensional Kinematics and Projectile Motion Concepts

• Projectile Motion Defined:

  • A projectile is any object moving through two-dimensional space that experiences no forces other than the downward force of gravity.

  • Air resistance is neglected in ideal projectile calculations.

• Independence of Vertical and Horizontal Motions:

  • Horizontal motion ($x$) and vertical motion ($y$) are completely independent orthogonal processes.

  • Horizontal Component ($x$): Experiences zero acceleration ($a_x = 0$). The horizontal velocity component remains strictly constant throughout the entire flight ($v_x = (v_x)_i = \text{constant}$).

  • Vertical Component ($y$): Experiences constant downward acceleration due to gravity ($a_y = -g = -9.80\,m/s^2$ or $-10\,m/s^2$).

  • A projectile launched horizontally with initial velocity $v_{x0}$ and an object dropped simultaneously from rest from the exact same height hit the ground at the exact same instant.

• Initial Velocity Breakdown for Angled Launches:

  • For a projectile launched at speed $v_i$ at a launch angle $\theta$ above the horizontal:         $(v_x)_i = v_i \cos(\theta)$         $(v_y)_i = v_i \sin(\theta)$

• Trajectory Characteristics and Symmetry:

  • The path (trajectory) followed by an ideal projectile is a parabola.

  • At Apex (Peak Height): The vertical velocity component momentarily equals zero ($v_y = 0$), but the overall speed is non-zero because $v_x = (v_x)_i$. Acceleration at the apex remains strictly $a_y = -g$ downward.

  • Symmetry of Flight: For a projectile landing at the same vertical height from which it was launched:

    • Time spent rising equals time spent falling ($t_{up} = t_{down} = \frac{1}{2} t_{total}$).

    • The speed $v$ at any given height during ascent equals the speed $v$ at that exact same height during descent.

    • The vertical velocity component at landing is equal in magnitude and opposite in direction to the initial vertical velocity ($(v_y)_f = -(v_y)_i$).

Mathematical Formulations and Derivations for Projectiles

• Governing Kinematic Equations for Projectile Motion:

  • Horizontal Dimension ($a_x = 0$):         $x_f = x_i + v_x \Delta t$         $v_x = (v_x)_i = \text{constant}$

  • Vertical Dimension ($a_y = -g$):         $y_f = y_i + (v_y)_i \Delta t - \frac{1}{2} g (\Delta t)^2$         $(v_y)_f = (v_y)_i - g \Delta t$         $(v_y)_f^2 = (v_y)_i^2 - 2g(y_f - y_i)$

• Coupling Variable: Time interval $\Delta t$ is the sole shared scalar parameter linking the independent horizontal and vertical kinematic equations.

• Derivations for Symmetric Level-Ground Projectiles ($y_f = y_i$):

  • Total Hang Time ($t_{total}$):         Set $\Delta y = 0$ in $\Delta y = (v_y)_i t - \frac{1}{2}g t^2$:         $0 = v_i \sin(\theta) t - \frac{1}{2}g t^2$         $t_{total} = \frac{2 v_i \sin(\theta)}{g}$

  • Maximum Height ($H_{max}$):         Set $v_y = 0$ at peak height in $v_y^2 = (v_y)_i^2 - 2g H_{max}$:         $0 = (v_i \sin(\theta))^2 - 2g H_{max}$         $H_{max} = \frac{v_i^2 \sin^2(\theta)}{2g}$

  • Horizontal Range Formula ($R$):         Substitute $t_{total}$ into $R = v_x t_{total}$:         $R = (v_i \cos(\theta)) \left(\frac{2 v_i \sin(\theta)}{g}\right) = \frac{v_i^2 (2 \sin(\theta) \cos(\theta))}{g}$         Using trigonometric identity $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$, the level-ground range formula simplifies to:         $R = \frac{v_i^2 \sin(2\theta)}{g}$

• Launch Angle Dependencies and Properties:

  • Maximum Range Angle: On flat terrain, a launch angle of $\theta = 45^\circ$ maximizes horizontal range because $\sin(2 \times 45^\circ) = \sin(90^\circ) = 1$.

  • Complementary Angles Property: Any two launch angles $\theta$ and $90^\circ - \theta$ (e.g., $15^\circ$ and $75^\circ$, or $30^\circ$ and $60^\circ$) shot at the exact same launch speed $v_i$ yield identical horizontal ranges $R$.

  • Trade-off Behavior: Larger launch angles produce greater maximum vertical heights $H_{max}$ and longer total hang times $t_{total}$, but smaller horizontal velocities $v_x$.

Comprehensive Conceptual Problems and Case Studies

• Moon Velocity Change Problem:

  • Scenario: The Moon orbits Earth counterclockwise. Position 1 is at 3 o'clock moving North ($\vec{v}_1$). Position 2 is at 12 o'clock moving West ($\vec{v}_2$).

  • To find change in velocity $\Delta \vec{v} = \vec{v}_2 - \vec{v}_1$:

    • Draw $\vec{v}_2$ pointing West (left).

    • Draw $-\vec{v}_1$ pointing South (down).

    • Resultant $\Delta \vec{v}$ points Down and to the Left (Southwest).

• Ice Gliders Average Acceleration Comparison:

  • Glider A: Changes velocity from $30\,m/s\text{ NW}$ to $20\,m/s\text{ NW}$ in $3\,s$.         $\Delta v_A = 20 - 30 = -10\,m/s\text{ NW}$         $a_A = \frac{|-10\,m/s|}{3\,s} = 3.33\,m/s^2$

  • Glider B: Changes velocity from $10\,m/s\text{ NW}$ to $10\,m/s\text{ SE}$ in $6\,s$.         $\Delta v_B = -10 - (10) = -20\,m/s\text{ NW}$         $a_B = \frac{|-20\,m/s|}{6\,s} = 3.33\,m/s^2$

  • Conclusion: The magnitudes of average acceleration for Glider A and Glider B are strictly equal.

• Speedboats Changing Velocities Misconception:

  • Statement: "Boat A ($15\,m/s\text{ East}$ to $20\,m/s\text{ North}$ in $10\,s$) and Boat B ($10\,m/s\text{ West}$ to $15\,m/s\text{ West}$ in $10\,s$) have identical accelerations because both change speed by $5\,m/s$".

  • Correction: The student is INCORRECT. Acceleration is a vector quantity ($\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$).

  • For Boat A: $\Delta \vec{v}_A = \sqrt{15^2 + 20^2} = 25\,m/s$, giving $a_A = \frac{25}{10} = 2.5\,m/s^2$.

  • For Boat B: $\Delta \vec{v}_B = 15 - 10 = 5\,m/s$, giving $a_B = \frac{5}{10} = 0.5\,m/s^2$

  • Boat A experiences a significantly larger acceleration magnitude because its direction changes orthogonally.

• Rocks Thrown Off Cliff Analysis:

  • Two identical rocks thrown horizontally from the same height at the same time: Rock A closer to cliff, Rock B farther out.

  • Horizontal velocity: Rock A is less than Rock B (Rock B traveled farther horizontally in the same time interval).

  • Vertical velocity: Rock A equals Rock B (both started with zero vertical velocity and accelerated downward at $g$).

  • Horizontal acceleration: Rock A equals Rock B (both equal $0\,m/s^2$).

  • Vertical acceleration: Rock A equals Rock B (both equal $-g = -9.80\,m/s^2$).

• Toy Trucks Rolling Off Identical Tables:

  • Truck A ($8\,m/s$, $500\,g$) vs. Truck B ($5\,m/s$, $800\,g$) rolling off identical $30\,cm$ high tables.

  • Time in air: Both trucks spend the EXACT SAME TIME in the air.

  • Reasoning: Fall time is governed exclusively by height $h$ and downward gravity acceleration $g$ ($t = \sqrt{\frac{2h}{g}}$). Fall time is completely independent of horizontal speed and object mass.

• Baseball Field Projectile Graphs:

  • Coordinate System: Origin at point S in right field, positive $y$ is UP, positive $x$ is RIGHT (toward home plate is LEFT/negative $x$).

  • Horizontal Velocity ($v_x$): Constant horizontal line located below the time axis (negative value).

  • Vertical Velocity ($v_y$): Straight linear line starting at positive value, crossing zero at peak height, extending to negative values.

  • Horizontal Acceleration ($a_x$): Flat horizontal line lying directly on the zero axis ($a_x = 0$).

  • Vertical Acceleration ($a_y$): Flat horizontal line below the time axis at constant value $-g$ ($a_y = -9.80\,m/s^2$).