Comprehensive Study Notes on Kinematics: One and Two-Dimensional Relative Motion

Fundamentals of Relative Motion in One Dimension

• Concept of Relative Velocity:

  • Motion is dependent on the reference frame of the observer.
  • An object's velocity measured by an observer on the ground differs from its velocity measured by an observer who is also in motion.

• One-Dimensional Cart and Ball Example:

  • A cart moves along a track in the forward direction at a velocity of $2\,m/s$ relative to the ground.
  • A person inside the cart tosses a ball forward at a velocity of $3\,m/s$ relative to themselves.
  • Forward Motion Calculation:
    • To an observer standing still on the ground, the velocity of the ball is the sum of the cart's velocity and the ball's velocity relative to the cart:             $v = 2\,m/s + 3\,m/s = 5\,m/s$
  • Airplane Analogy:
    • In an airplane flying forward at $600\,mph$, tossing a ball forward down the aisle at $3\,mph$ relative to the passenger results in a ground speed of $603\,mph$.
  • Backward Motion Calculation:
    • If the person on the cart tosses the ball backward (in the negative direction) at $3\,m/s$ relative to the cart:
      • Cart velocity relative to ground ($+x$ direction): $v_{cart/ground} = +2\,m/s$
      • Ball velocity relative to cart ($-x$ direction): $v_{ball/cart} = -3\,m/s$
      • Ball velocity relative to ground observer:                 $v_{ball/ground} = v_{ball/cart} + v_{cart/ground} = -3\,m/s + 2\,m/s = -1\,m/s$
    • The ground observer sees the ball moving backward at $1\,m/s$.

Frame of Reference Vector Equations and Subscript Rules

• General Addition Equation for Relative Motion:

  • Relative velocity addition is fundamentally a vector addition problem.
  • The basic relative motion equation relating three reference frames ($A$, $B$, and $C$) is given by:         $v_{A/B} = v_{A/C} + v_{C/B}$
  • Subscript Matching Convention:
    • The first subscript of the left side ($A$) matches the first subscript of the first term on the right side ($A$).
    • The second subscript of the left side ($B$) matches the second subscript of the second term on the right side ($B$).
    • The inner subscripts ($C$) must match each other and represent the intermediate reference frame.

• Extended Reference Frame Chains:

  • The subscript addition rule can be extended to an arbitrary number of intermediate frames:         $v_{A/D} = v_{A/B} + v_{B/C} + v_{C/D}$
  • In one-dimensional motion, vector addition is managed by strictly maintaining positive and negative signs relative to a chosen $x$-axis.

• Inverse Reference Frame Identity:

  • Switching the observer frame inverts the direction (sign) of the velocity vector:         $v_{A/B} = -v_{B/A}$
  • If Frame $A$ moves at $+12\,m/s$ relative to Frame $B$, then Frame $B$ moves at $-12\,m/s$ relative to Frame $A$.

Speed versus Velocity and Sign Conventions

• Distinction Between Speed and Velocity:

  • Speed: A scalar quantity that is strictly non-negative. Automobiles feature speedometers that display speed, which cannot register negative values.
  • Velocity: A vector quantity that incorporates both magnitude (speed) and direction. Velocity can be positive or negative based on the coordinate system.

• Sign Convention Protocol:

  • Always establish a coordinate system (e.g., standard $x$-axis pointing right/East as positive) before setting up equations.
  • Motion in the positive direction yields positive velocity.
  • Motion in the opposite direction yields negative velocity.

Opposing Train Motion Analysis

• Red Line Opposing Trains Example:
  • Two trains move in opposite directions along a straight track:
    • Person 1 (in Train 1) moves North at $25\,m/s$ relative to the ground.
    • Person 2 (in Train 2) moves South at $35\,m/s$ relative to the ground.
  • Coordinate Setup:
    • Define North as the positive $x$-direction ($+x$).
    • Velocity of Person 1 relative to ground: $v_{1/G} = +25\,m/s$
    • Velocity of Person 2 relative to ground: $v_{2/G} = -35\,m/s$
  • Calculating Velocity of Person 2 Relative to Person 1 ($v_{2/1}$):
    • Using the relative motion equation:             $v_{2/1} = v_{2/G} + v_{G/1}$
    • Applying the inverse identity $v_{G/1} = -v_{1/G}$, the formula becomes:             $v_{2/1} = v_{2/G} - v_{1/G}$
    • Substituting values:             $v_{2/1} = -35\,m/s - 25\,m/s = -60\,m/s$
    • Person 1 observes Person 2 approaching in the negative direction at 60\,m/s$.\n * **Calculating Velocity of Person 1 Relative to Person 2 (v_{1/2}):**\n * v_{1/2} = -v_{2/1} = -(-60\,m/s) = +60\,m/s\n * Person 2 observes Person 1 approaching in the positive direction at 60\,m/s$.
  • Physical Implications:
    • When two objects move toward each other, their relative speed of approach is the sum of their individual speeds relative to the ground ($25\,m/s + 35\,m/s = 60\,m/s$).
    • Impacting an oncoming moving wall creates a much greater relative impact speed than impacting a stationary wall.

Inertial Reference Frames and Projectile Equivalence

• Inertial Reference Frames:

  • An inertial frame is any reference frame moving at a constant velocity (zero acceleration).
  • The fundamental laws of physics are identical in all inertial reference frames.
  • Observations can be fully analyzed in any chosen reference frame; adjustments between frames require applying the relative velocity formula.

• Train Vertical Toss Analysis:

  • Inside Frame (Person on train moving at constant velocity): Tossing a ball straight up results in simple one-dimensional vertical motion straight up and down.
  • Outside Frame (Observer on ground): The ball possesses both initial vertical velocity and horizontal forward velocity matching the train, creating a two-dimensional parabolic projectile motion trajectory.

• Mike on the Metro Example:

  • Mike walks forward at $1\,m/s$ inside a Metro car.
  • The Metro car travels forward at $30\,m/s$ relative to the ground.
  • Mike's velocity relative to the ground is:         $v_{M/G} = v_{M/Metro} + v_{Metro/G} = +1\,m/s + 30\,m/s = +31\,m/s$

• Unit Conversion Estimation Rule of Thumb:

  • To convert meters per second ($m/s$) to miles per hour ($mph$):
    • Double the numerical value in $m/s$ and add approximately $10\%$.
    • Example: 30\,m/s \n        \rightarrow (30 \times 2) + 10\% = 60 + 6 = 66\,mph.

Walkway Kinematics: Woman and Dog Scenario

• Problem Setup:

  • A moving walkway at an airport moves to the right ($+x$ direction) at $2\,m/s$ relative to the ground.
  • A woman stands stationary on the walkway ($v_{W/Belt} = 0\,m/s$).
  • A dog runs to the left ($-x$ direction) along the walkway at $8\,m/s$ relative to the walkway.
  • Coordinate System: Rightward motion is positive ($+x$), leftward motion is negative ($-x$).

• Calculation 1: Dog Velocity Relative to Ground ($v_{D/G}$):

  • Equation:         $v_{D/G} = v_{D/Belt} + v_{Belt/G}$
  • Substituting values:         $v_{D/G} = -8\,m/s + 2\,m/s = -6\,m/s$
  • The dog moves left relative to the ground with a velocity of $-6\,m/s$ (a ground speed of $6\,m/s$).

• Calculation 2: Dog Velocity Relative to the Woman ($v_{D/W}$):

  • Equation:         $v_{D/W} = v_{D/Belt} + v_{Belt/W}$
  • Since the woman is standing still on the belt, $v_{Belt/W} = 0\,m/s$
  • Substituting values:         $v_{D/W} = -8\,m/s + 0\,m/s = -8\,m/s$
  • Because both the dog and the woman inhabit the reference frame of the belt ("their world"), the belt's motion relative to the ground drops out entirely.

Graphing Relative Position and Two-Ball Collision Analysis

• Problem Setup:

  • Two balls move along a one-dimensional path:
    • Green ball ($G$) moves East ($+x$ direction) at $v_{G} = +6\,m/s$ relative to the world.
    • Red ball ($R$) moves West ($-x$ direction) at $v_{R} = -2\,m/s$ relative to the world.
  • Initial distance between the two balls at $t = 0\,s$ is $12\,m$ (Green ball is behind/West of the Red ball).

• Constructing Frame of Reference centered on the Red Ball:

  • Set the Red ball as the origin of the coordinate system ($x_R = 0$).
  • Initial Relative Position ($x_{G/R,0}$):
    • Since the Green ball is $12\,m$ West (behind) the Red ball, its initial relative position is:             $x_{G/R,0} = -12\,m$
  • Relative Velocity Calculation ($v_{G/R}$):
    • Equation:             $v_{G/R} = v_{G/World} + v_{World/R} = v_{G/World} - v_{R/World}$
    • Substituting values:             $v_{G/R} = +6\,m/s - (-2\,m/s) = +8\,m/s$
    • From the Red ball's perspective, the Green ball approaches it with a positive velocity of +8\,m/s$.\n\n* **Relative Position Equation of Motion:**\n * x_{G/R}(t) = x_{G/R,0} + v_{G/R} t\n * x_{G/R}(t) = -12 + 8t\n\n* **Graph Characteristics (Position x_{G/R}$vs. Time$t):**\n * **Vertical Intercept (t = 0$):**$-12\,m\n * **Slope:** +8\,m/s\n * **Collision Point (x_{G/R} = 0):**\n        0 = -12 + 8t \rightarrow 8t = 12 \rightarrow t = \frac{12}{8} = 1.5\,s\n * The graph is a straight line starting at -12\,m$at$t = 0\,s$and increasing linearly to$0\,m$at$t = 1.5\,s.\n\n* **Symmetry Note:**\n * The position graph of the Red ball relative to the Green ball (x_{R/G}$) is the inverted mirror image: starting at$+12\,m$with a slope of$-8\,m/s$, hitting$0\,m$at$t = 1.5\,s\n\n# Two-Dimensional Relative Motion: River Currents and Riptide Strategy\n\n* **Practical Safety Application (Riptides and River Crossing):**\n * When caught in a water current or riptide directed parallel/away from the shore:\n * Fighting the current head-on leads to exhaustion and drowning.\n * The optimal survival technique is swimming straight toward the shore (y$-direction perpendicular to current), completely ignoring sideways drift ($x-direction).\n * Once safety (the bank/sand) is reached, walking back to the original target point on land requires far less energy than swimming against water currents.\n\n* **Feasibility Analysis of Direct Perpendicular Crossing:**\n * **Scenario Setup:**\n * Swimmer velocity relative to water: v_{S/W} = 0.4\,m/s\n * Water current velocity relative to ground: v_{W/G} = 0.5\,m/s (directed downstream/rightward).\n * Target Point X is located directly across the river, perpendicular to the bank.\n * **Vector Analysis:**\n * The swimmer's ground velocity vector is:\n            \mathbf{v}{S/G} = \mathbf{v}{S/W} + \mathbf{v}{W/G}\n * To swim directly across to Point X$, the downstream current ($0.5\,m/s) must be completely canceled by an upstream horizontal vector component of the swimmer's effort.\n * Maximum upstream vector component the swimmer can produce is |\mathbf{v}{S/W}| = 0.4\,m/s.\n * Because 0.4\,m/s < 0.5\,m/s, the upstream effort can never balance the downstream current.\n * **Conclusion:** It is physically impossible for the swimmer to reach Point X directly, regardless of the angle angled upstream.\n\n# River Crossing Race: The Triplets Strategy Comparison\n\n* **Problem Setup:**\n * Triplets Anne, Beth, and Curly have identical swimming capabilities, achieving identical maximum swimming speeds relative to the water (v_{S/W}).\n * They compete to see who can reach the opposite river bank first.\n * **Strategies Employed:**\n * **Anne:** Angled 30^\circ upstream relative to the perpendicular line to compensate for river drift.\n * **Curly:** Angled 30^\circ downstream relative to the perpendicular line to let the current assist speed.\n * **Beth:** Swims straight across, aiming directly perpendicular to the opposite bank.\n\n* **Mathematical Derivation of Crossing Time:**\n * Let the y$-axis point straight across the river (width =$D$) perpendicular to the banks, and the$x -axis point downstream parallel to the banks.\n * The time required to cross the river (t_{cross}$) is strictly determined by the perpendicular ($y$) component of ground velocity ($v_y):\n        t_{cross} = \frac{D}{v_y}\n\n* **Velocity Component Analysis:**\n * **Beth:** Swims entirely along the y-axis:\n        v_{y, Beth} = v_{S/W}\n * **Anne:** Angles 30^\circ upstream:\n        v_{y, Anne} = v_{S/W} \cos(30^\circ) \approx 0.866 \cdot v_{S/W}\n * **Curly:** Angles 30^\circ downstream:\n        v_{y, Curly} = v_{S/W} \cos(30^\circ) \approx 0.866 \cdot v_{S/W}\n\n* **Race Results and Conclusions:**\n * **First to Cross:** **Beth wins the race** because v_{y, Beth}$is maximized, making her crossing time$t_{cross}$$ the shortest.
  • Subsequent Order: Anne and Curly tie for second place, taking significantly longer than Beth to reach the opposite bank.
  • Landing Positions:
    • Anne lands furthest upstream relative to her heading.
    • Beth lands downstream from the perpendicular point.
    • Curly lands furthest downstream.

Course Organization and Examination Scope

• Kinematics Unit Scope:

  • Relative motion represents the final lecture topic within the Kinematics module.

• Exam 1 Content Boundaries:

  • Included on Exam 1: Relative Motion and Newton's Laws of Motion.
  • Excluded from Exam 1: Rotational and Circular Motion (deferred to subsequent lectures covering forces and dynamics).