Two-Dimensional Kinematics, Vector Addition, Projectile Motion, and Relative Velocity

Kinematics in Two Dimensions and Independence of Motion

• Motion in two dimensions consists of separate horizontal and vertical components that act simultaneously along perpendicular directions.

• Navigating a two-dimensional grid:

  • Pedestrians and vehicles in a grid-like street layout, such as New York, cannot travel in direct straight lines and must follow perpendicular roads and sidewalks.

  • Walking $14.0\,\text{blocks}$ in total along a city grid consisting of $9.00\,\text{blocks}$ east followed by $5.00\,\text{blocks}$ north forms a right triangle.

  • Straight-line distance calculation via the Pythagorean theorem:

  • $a^2 + b^2 = c^2$

  • $d = \sqrt{(9.00\,\text{blocks})^2 + (5.00\,\text{blocks})^2} = 10.3\,\text{blocks}$

  • The straight-line displacement of $10.3\,\text{blocks}$ is substantially shorter than the $14.0\,\text{blocks}$ walked along the grid.

  • Significant figures convention: Discrete block measurements like $9$ and $5$ represent exact discrete values (treated as $9.00$ and $5.00$) to yield answers expressed precisely to three significant figures.

• Core principle of the independence of perpendicular motions:

  • Horizontal and vertical components of two-dimensional motion are completely independent of each other.

  • Motion in the horizontal direction has no effect on motion in the vertical direction, and vertical motion has no effect on horizontal motion.

  • Walking east is affected solely by eastward motion; walking north is affected solely by northward motion.

• Experimental evidence using two baseballs:

  • One baseball is dropped vertically from rest from a specified height.

  • A second baseball is thrown horizontally from the exact same height at the exact same instant, tracing a curved two-dimensional path.

  • Stroboscopic photography capturing the falling baseballs at fixed time intervals demonstrates that the vertical positions of both balls are identical at every strobe flash.

  • Under ideal conditions neglecting air resistance, the vertical motion of any falling projectile is governed strictly by gravity and remains completely unaffected by horizontal motion.

  • The horizontally thrown ball travels equal horizontal distances during each fixed time interval between strobe flashes because no horizontal forces act on it after launch.

  • Horizontal velocity remains constant and is unaffected by gravity or vertical velocity.

  • Real-world caveat: Atmospheric air resistance introduces drag forces that alter the velocities of moving bodies in both dimensions.

Vector Representation and Graphical Addition Methods

• Fundamental vector properties:

  • A vector is a quantity characterized by both magnitude and direction, such as displacement, velocity, acceleration, and force.

  • Scalar quantities possess magnitude only.

  • Vector notation: Vectors are designated by boldface symbols (e.g., $\mathbf{d}$, $\mathbf{F}$), vector magnitudes are indicated by italicized symbols (e.g., $d$, $F$), and direction is specified relative to coordinate axes via an angle $\theta$.

  • Graphic representation: An arrow represents a vector; the arrow length is directly proportional to vector magnitude, hash marks indicate scale units, and the arrowhead defines direction.

  • Total two-dimensional displacement is represented by three vectors: the horizontal component vector, the vertical component vector, and the resultant straight-line displacement vector connecting initial and final positions.

• Head-to-tail method for graphical vector addition:

  • The tail represents the initial starting point of the vector; the head (or tip) represents the pointed arrow end.

  • Step 1: Draw the first vector as an arrow from the origin, scaling its length with a ruler and measuring its angle relative to the reference frame with a protractor (e.g., $9.00\,\text{blocks}$ east).

  • Step 2: Draw the second vector starting with its tail placed directly at the head of the first vector (e.g., $5.00\,\text{blocks}$ north).

  • Step 3: For systems with more than two vectors, continue attaching each subsequent vector tail to the head of the preceding vector.

  • Step 4: Draw a resultant arrow $\mathbf{R}$ originating from the tail of the first vector to the head of the final vector.

  • Step 5: Measure the magnitude of the resultant vector directly with a ruler using the established map or drawing scale.

  • Step 6: Measure the direction angle of the resultant vector relative to the reference axis using a protractor.

  • Accuracy limitations: Graphical addition precision is governed strictly by drawing scale precision and measuring instrument quality.

• Commutative property of vector addition:

  • Vector addition is independent of the order in which vectors are added:

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

  • This commutative behavior is identical to ordinary scalar arithmetic ($2 + 3 = 3 + 2$).

• Multi-step flat-field displacement example:

  • A pedestrian traverses three sequential path segments across a flat field:

  • Vector $\mathbf{A}$: $25.0\,m$ directed $49.0^\circ$ north of east.

  • Vector $\mathbf{B}$: $23.0\,m$ directed $15.0^\circ$ north of east.

  • Vector $\mathbf{C}$: $32.0\,m$ directed $68.0^\circ$ south of east.

  • By placing vectors head-to-tail and drawing resultant vector $\mathbf{R}$ from the origin to the head of $\mathbf{C}$, direct measurement yields:

  • Resultant magnitude: $R = 50.8\,m$

  • Resultant direction: $\theta = 5.47^\circ$ south of east.

Vector Subtraction and Scalar Multiplication

• Graphical vector subtraction:

  • The negative of a vector $\mathbf{B}$, denoted $-\mathbf{B}$, is defined as a vector having the exact same magnitude as $\mathbf{B}$ but pointing in the opposite direction ($180^\circ$ reversal).

  • Vector subtraction is mathematically defined as the addition of the corresponding negative vector:

  • $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$

  • Subtraction order is analogous to scalar subtraction ($5 - 2 = 5 + (-2)$).

• Boat navigation error scenario:

  • A boat navigator intends to reach a dock using two legs:

  • Intended Leg 1 ($\mathbf{A}$): $27.5\,m$ at $66.0^\circ$ north of east.

  • Intended Leg 2 ($\mathbf{B}$): $30.0\,m$ at $112^\circ$ north of east (equivalent to $22.0^\circ$ west of north).

  • Actual correct dock location ($\mathbf{A} + \mathbf{B}$): $52.9\,m$ at $90.1^\circ$ north of east.

  • Erroneous course execution: The navigator travels in the exact opposite direction on the second leg:

  • Traveled vector $-\mathbf{B}$: $30.0\,m$ directed along $180^\circ - 112^\circ = 68.0^\circ$ south of east.

  • Accidental ending point ($\mathbf{A} - \mathbf{B}$): $23.0\,m$ at $7.5^\circ$ south of east.

  • The navigator ends up at a location significantly displaced from the intended destination.

• Multiplication and division of vectors by scalars:

  • Multiplying a vector $\mathbf{A}$ by a scalar $c$ scales the vector magnitude to the product value:

  • $\text{Magnitude} = |c|A$

  • If $c > 0$, the vector direction is unchanged.

  • If $c < 0$, the magnitude scales by $|c|$ and the vector direction reverses by $180^\circ$.

  • Example: Walking three times farther on a $27.5\,m$ course at $66.0^\circ$ north of east yields $3 \times 27.5\,m = 82.5\,m$ along $66.0^\circ$ north of east.

  • Multiplying by $-2$ doubles the vector magnitude and reverses its directional orientation.

  • Division of a vector by a scalar is equivalent to multiplying the vector by the reciprocal scalar value $\frac{1}{c}$.

Analytical Methods for Vector Components and Addition

• Comparison of graphical and analytical methods:

  • Graphical methods rely on rulers and protractors, constrained by drafting precision.

  • Analytical methods use geometry and trigonometry, limited only by the known accuracy and precision of the underlying physical values.

• Resolving a vector into perpendicular components:

  • Any two-dimensional vector $\mathbf{A}$ positioned on a Cartesian coordinate plane can be resolved into perpendicular components $\mathbf{A}_x$ and $\mathbf{A}_y$ along the $x$-axis and $y$-axis:

  • $\mathbf{A}_x + \mathbf{A}_y = \mathbf{A}$

  • Warning: Vector equality does not denote algebraic summation of scalar magnitudes. For example, a $3.00\,m$ eastward vector added to a $4.00\,m$ northward vector produces a resultant displacement magnitude of $5.00\,m$, whereas $3.00 + 4.00 = 7.00 \neq 5.00$.

  • Trigonometric decomposition of vector components:

  • $A_x = A\cos(\theta)$

  • $A_y = A\sin(\theta)$

  • Example: A total displacement vector of $A = 10.3\,\text{blocks}$ oriented at $\theta = 29.1^\circ$ north of east resolves into:

  • $A_x = (10.3\,\text{blocks})\cos(29.1^\circ) = 9.00\,\text{blocks}$

  • $A_y = (10.3\,\text{blocks})\sin(29.1^\circ) = 5.00\,\text{blocks}$

• Calculating a resultant vector from known perpendicular components:

  • Magnitude determination via the Pythagorean theorem:

  • $A = \sqrt{A_x^2 + A_y^2}$

  • Direction angle determination via the inverse tangent function:

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

  • Example verification: For components $A_x = 9.00\,\text{blocks}$ and $A_y = 5.00\,\text{blocks}$:

  • $A = \sqrt{(9.00)^2 + (5.00)^2} = 10.3\,\text{blocks}$

  • $\theta = \tan^{-1}\left(\frac{5.00}{9.00}\right) = 29.1^\circ$

• Step-by-step procedure for analytical vector addition:

  • Step 1: Establish coordinate axes ($x$ horizontal, $y$ vertical). Resolve each vector into components along these axes using $A_x = A\cos(\theta_A)$, $A_y = A\sin(\theta_A)$, $B_x = B\cos(\theta_B)$, and $B_y = B\sin(\theta_B)$.

  • Step 2: Sum component magnitudes algebraically along each axis to obtain resultant components:

  • $R_x = A_x + B_x$

  • $R_y = A_y + B_y$

  • Step 3: Compute the magnitude of resultant vector $\mathbf{R}$:

  • $R = \sqrt{R_x^2 + R_y^2}$

  • Step 4: Compute the direction angle of resultant vector $\mathbf{R}$ relative to the horizontal axis:

  • $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

• Worked example of analytical vector addition:

  • Vector $\mathbf{A}$: Magnitude $A = 53.0\,m$, angle $\theta_A = 20.0^\circ$ north of east.

  • $A_x = (53.0\,m)\cos(20.0^\circ) = (53.0\,m)(0.940) = 49.8\,m$

  • $A_y = (53.0\,m)\sin(20.0^\circ) = (53.0\,m)(0.342) = 18.1\,m$

  • Vector $\mathbf{B}$: Magnitude $B = 34.0\,m$, angle $\theta_B = 63.0^\circ$ north of east.

  • $B_x = (34.0\,m)\cos(63.0^\circ) = (34.0\,m)(0.454) = 15.4\,m$

  • $B_y = (34.0\,m)\sin(63.0^\circ) = (34.0\,m)(0.891) = 30.3\,m$

  • Summation of components:

  • $R_x = 49.8\,m + 15.4\,m = 65.2\,m$

  • $R_y = 18.1\,m + 30.3\,m = 48.4\,m$

  • Resultant magnitude and direction calculation:

  • $R = \sqrt{(65.2\,m)^2 + (48.4\,m)^2} = 81.2\,m$

  • $\theta = \tan^{-1}\left(\frac{48.4\,m}{65.2\,m}\right) = \tan^{-1}(0.742) = 36.6^\circ$

  • The resultant displacement is $81.2\,m$ directed at $36.6^\circ$ north of east.

• Analytical vector subtraction:

  • Vector subtraction replaces the components of $\mathbf{B}$ with their negative values:

  • $R_x = A_x - B_x$

  • $R_y = A_y - B_y$

  • Magnitude and angle equations remain identical following component subtraction.

Two-Dimensional Projectile Motion Fundamentals

• Definitions:

  • Projectile motion: The motion of an object thrown or projected into the air, subject solely to the acceleration due to gravity.

  • Projectile: The moving object traveling through space.

  • Trajectory: The curved path traced by a projectile through space.

• Physical conditions and coordinate conventions:

  • Air resistance, friction, and secondary aerodynamic effects are assumed to be negligible.

  • Acceleration components:

  • Horizontal acceleration: $a_x = 0\,m/s^2$

  • Vertical acceleration: $a_y = -g = -9.80\,m/s^2$ (assuming upward is positive)

  • If downward is defined as positive, $a_y = +g = +9.80\,m/s^2$.

  • Initial velocity vector decomposition:

  • $v_{0x} = v_0\cos(\theta_0)$

  • $v_{0y} = v_0\sin(\theta_0)$

• Kinematic equations for projectile motion:

  • Horizontal motion equations (constant velocity):

  • $a_x = 0\,m/s^2$

  • $v_x = v_{0x} = \text{constant}$

  • $x = x_0 + v_x t$

  • Vertical motion equations (constant acceleration):

  • $a_y = -g = -9.80\,m/s^2$

  • $v_y = v_{0y} - gt$

  • $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$

  • $v_y^2 = v_{0y}^2 - 2g(y - y_0)$

  • $y = y_0 + \frac{1}{2}(v_{0y} + v_y)t$

  • Common parameter: Time $t$ is the single scalar quantity shared by both the horizontal and vertical motions.

• Recombining horizontal and vertical components:

  • Total displacement magnitude:

  • $s = \sqrt{x^2 + y^2}$

  • Displacement direction:

  • $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

  • Instantaneous velocity magnitude:

  • $v = \sqrt{v_x^2 + v_y^2}$

  • Instantaneous velocity direction:

  • $\theta_v = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Projectile Motion Scenarios and Worked Calculations

• Fireworks shell trajectory analysis:

  • Launch initial conditions:

  • Initial speed: $v_0 = 70.0\,m/s$

  • Launch angle: $\theta_0 = 75.0^\circ$ above horizontal

  • Fuse timed to detonate the shell precisely at its trajectory apex ($v_y = 0\,m/s$)

  • Initial positions: $x_0 = 0\,m$, $y_0 = 0\,m$

  • Initial component velocities:

  • $v_{0x} = (70.0\,m/s)\cos(75.0^\circ) = 18.1\,m/s$

  • $v_{0y} = (70.0\,m/s)\sin(75.0^\circ) = 67.6\,m/s$

  • Maximum vertical altitude ($y$) at detonation:

  • $v_y^2 = v_{0y}^2 - 2gy$

  • $0 = (67.6\,m/s)^2 - 2(9.80\,m/s^2)y$

  • $y = \frac{(67.6\,m/s)^2}{2(9.80\,m/s^2)} = 233\,m$

  • General maximum height formula for any projectile:

  • $h = \frac{v_{0y}^2}{2g}$

  • Time elapsed ($t$) from launch to explosion:

  • $y = \frac{1}{2}(v_{0y} + v_y)t$

  • $233\,m = \frac{1}{2}(67.6\,m/s + 0\,m/s)t$

  • $t = \frac{2(233\,m)}{67.6\,m/s} = 6.90\,s$

  • Alternative time check: $v_y = v_{0y} - gt \implies 0 = 67.6 - 9.80t \implies t = 6.90\,s$

  • Horizontal displacement ($x$) at detonation:

  • $x = v_x t = (18.1\,m/s)(6.90\,s) = 125\,m$

• Kilauea volcano ejected rock projectile:

  • Launch parameters:

  • Ejection speed: $v_0 = 25.0\,m/s$

  • Ejection angle: $\theta_0 = 35.0^\circ$ above horizontal

  • Impact altitude: $20.0\,m$ below launch level ($y = -20.0\,m$ relative to $y_0 = 0\,m$)

  • Velocity components at launch:

  • $v_{0x} = (25.0\,m/s)\cos(35.0^\circ) = 20.5\,m/s$

  • $v_{0y} = (25.0\,m/s)\sin(35.0^\circ) = 14.3\,m/s$

  • Total flight time determination:

  • $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$

  • $-20.0\,m = 0 + (14.3\,m/s)t - (4.90\,m/s^2)t^2$

  • $(4.90\,m/s^2)t^2 - (14.3\,m/s)t - 20.0\,m = 0$

  • Applying the quadratic formula for $at^2 + bt + c = 0$ with $a = 4.90$, $b = -14.3$, $c = -20.0$:

    • $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

    • Yields roots $t = 3.96\,s$ and $t = -1.03\,s$

  • Discarding the unphysical negative time corresponding to pre-launch trajectory yields $t = 3.96\,s$.

  • Final velocity components at impact:

  • Horizontal velocity: $v_x = v_{0x} = 20.5\,m/s$

  • Vertical velocity: $v_y = v_{0y} - gt = 14.3\,m/s - (9.80\,m/s^2)(3.96\,s) = -24.5\,m/s$

  • Impact velocity magnitude:

  • $v = \sqrt{v_x^2 + v_y^2} = \sqrt{(20.5\,m/s)^2 + (-24.5\,m/s)^2} = 31.9\,m/s$

  • Impact velocity angle:

  • $\theta_v = \tan^{-1}\left(\frac{v_y}{v_x}\right) = \tan^{-1}\left(\frac{-24.5\,m/s}{20.5\,m/s}\right) = \tan^{-1}(-1.19) = -50.1^\circ$

  • The negative angle indicates an orientation $50.1^\circ$ below the horizontal.

Projectile Range, Trajectory Symmetry, and Satellite Orbits

• Level-ground range properties:

  • Range ($R$) is defined as the horizontal distance traveled by a projectile on flat, level ground when returning to its initial release height.

  • Historical origin: Galileo Galilei first formulated the independence of perpendicular motions to predict projectile range for artillery and cannon aiming.

  • Level-ground range formula (neglecting air resistance):

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

• Launch angle dependencies:

  • For a fixed initial launch speed $v_0$, maximum range occurs at a launch angle of $\theta_0 = 45.0^\circ$, because $\sin(2 \times 45.0^\circ) = \sin(90.0^\circ) = 1$.

  • When atmospheric air resistance is included, the optimal launch angle for maximum range decreases to approximately $38.0^\circ$.

  • Angle symmetry: For any initial launch angle other than $45.0^\circ$, there exist two distinct launch angles that yield the exact same horizontal range.

  • The sum of these two complementary launch angles is exactly $90.0^\circ$ (for example, launch angles of $15.0^\circ$ and $75.0^\circ$ yield identical horizontal ranges, although their maximum altitudes and flight times differ markedly).

• Effect of gravitational acceleration on range:

  • Range is inversely proportional to local gravitational acceleration $g$.

  • Example: Lunar astronaut Alan Shepard drove a golf ball exceptionally large distances across the surface of the Moon due to the Moon's significantly weaker surface gravity.

• Trajectory scaling, Earth curvature, and orbital mechanics:

  • The standard range equation assumes a flat Earth where $R$ is tiny relative to Earth's circumference.

  • Over extremely long distances, Earth's spherical surface curves downward away from underneath the moving projectile.

  • As a consequence, the projectile has a greater vertical distance to fall than it would over flat ground, extending its actual range beyond standard equation predictions.

  • Orbital threshold: If a projectile is fired from a high altitude at a sufficiently high initial speed, the rate at which it falls due to gravity precisely matches the rate at which the surface of Earth curves away beneath it.

  • In orbital motion, the object continuously falls toward Earth without ever colliding with the surface, achieving continuous closed orbit.

Relative Velocity and Vector Addition of Velocities

• Relative velocity fundamentals:

  • Velocity measurements always depend on the frame of reference of the observer.

  • An object moving through a medium (e.g., water, air) possesses a velocity relative to the medium, while the medium itself possesses a velocity relative to an observer on solid ground.

  • The velocity of the object relative to the ground observer equals the vector sum of these velocities:

  • $\mathbf{v}_{\text{object, observer}} = \mathbf{v}_{\text{object, medium}} + \mathbf{v}_{\text{medium, observer}}$

• Physical examples of relative velocity:

  • Boat on a flowing river: A person rowing a boat directly perpendicular toward the opposite bank travels along a diagonal resultant path relative to a shore observer because the river current transports the boat downstream.

  • Airplane in a crosswind: An airplane flying forward relative to the surrounding air mass drifts sideways relative to the ground when subjected to a perpendicular crosswind.

• One-dimensional velocity addition:

  • Velocities along a common axis add as ordinary signed scalars.

  • Example: A field hockey player sprinting forward at $5.00\,m/s$ strikes a ball forward at $30.0\,m/s$ relative to her own motion. The ball's velocity relative to a stationary goalkeeper is:

  • $v = 5.00\,m/s + 30.0\,m/s = 35.0\,m/s$

• Two-dimensional velocity addition techniques:

  • Velocities in two dimensions are added using graphical or analytical vector methods.

  • Coordinate optimization: Mathematical calculations are significantly simplified by aligning one coordinate axis parallel to one of the velocity vectors (such as parallel to the river bank or parallel to gravity).