Notes on Motion in Two Dimensions

Chapter 4: Motion in Two Dimensions

Introduction to Motion in Two Dimensions

  • Using + or – signs is not always sufficient to fully describe motion in more than one dimension.
  • Vectors can provide a complete description of motion in multiple dimensions.
  • The study will focus on the vector nature of displacement, velocity, and acceleration.
  • These concepts will form the basis for multiple types of motion discussed in future chapters.

Position and Displacement

  • Position: Described by a position vector.
  • Displacement: Defined as the change in position, expressed between two points \
    • Initial position: $(x_0, y_0)$
    • Final position: $(x_f, y_f)$

General Motion Ideas

  • In two- or three-dimensional kinematics, the principles apply similarly to one-dimensional motion except full vector notation must be employed.
  • Positive and negative signs alone cannot sufficiently determine direction.

Average Velocity

  • Definition: The average velocity is the ratio of the displacement ($ ext{d}$) to the time interval ($ ext{t}$) for that displacement.
  • Equation: ext{Average Velocity} = rac{ ext{Displacement}}{ ext{Time Interval}}
  • The direction of the average velocity aligns with the displacement vector.
  • Average velocity depends only on initial and final positions, not on the path taken between them.

Instantaneous Velocity

  • Definition: Instantaneous velocity is the limit of average velocity as $ riangle t$ approaches zero.
  • As the time interval becomes smaller, the direction of displacement approaches the tangent to the curve at that point.
    • Magnitude: Absolute speed (a scalar quantity).

Average Acceleration

  • Definition: The average acceleration of a particle is defined as the change in the instantaneous velocity vector divided by the time interval during which the change occurs.
  • Equation: ext{Average Acceleration} = rac{ riangle ext{Velocity}}{ riangle t}
  • The direction of the average acceleration vector is found by vector subtraction.

Instantaneous Acceleration

  • Definition: Instantaneous acceleration is the limiting value of average acceleration as $ riangle t$ approaches zero.
  • Instantaneous acceleration equals the derivative of the velocity vector with respect to time.
  • Equation: ext{Instantaneous Acceleration} = rac{d ext{Velocity}}{dt}

Producing An Acceleration

  • Changes in a particle's motion can produce acceleration in several ways:
    • Magnitude of the velocity vector may change.
    • Direction of the velocity vector may change.
    • Both magnitude and direction can change simultaneously.

Kinematic Equations for Two-Dimensional Motion

  • When the two-dimensional motion experiences constant acceleration, a set of equations describes the motion, analogous to one-dimensional kinematics.
  • Motion can be modeled as two independent motions along the x and y axes; influences in the y direction do not affect motion in the x direction.
Position and Velocity Vectors
  • Position Vector: $( ext{x}, ext{y})$ in the xy-plane.
  • Velocity Vector: Can be derived from the position vector.
  • If acceleration is constant, the velocity vector can be expressed as:
    • Equation: extVelocity=extInitialVelocity+extAccelerationimesextTimeext{Velocity} = ext{Initial Velocity} + ext{Acceleration} imes ext{Time}
Position as Function of Time
  • The position vector can also be expressed as a function of time, indicating that it is the sum of:
    • Initial position vector.
    • The displacement from initial velocity.
    • The displacement from acceleration.
Graphical Representation of Final Velocity
  • Velocity vectors can be expressed through their components.
  • The resultant velocity vector is generally not aligned with the direction of individual axes.
Graphical Representation of Final Position
  • The vector for the position is also represented as components.
  • Individual components may not be directed along the same path as the resultant.
Summary of Graphical Representation
  • Relationships exist between changes in position or velocity and their respective effects on one another.

Projectile Motion

  • Defined as the motion of an object moving simultaneously in both x (horizontal) and y (vertical) directions.
Assumptions of Projectile Motion
  • Free-fall acceleration is constant and directed downward.
  • Assumes a flat Earth over the motion range, reasonable for small ranges compared to Earth's radius.
  • Negligible air friction leads objects in projectile motion to follow a parabolic path, called the trajectory.

Analyzing Projectile Motion

  • The motion is perceived as the superposition of motions in x and y directions:
    • Initial velocity: can be broken into components:
    • vxi=viimesextcos(heta)v_{xi} = v_i imes ext{cos}( heta)
    • vyi=viimesextsin(heta)v_{yi} = v_i imes ext{sin}( heta)
    • X-direction: Constant velocity, $ ext{a}_x = 0$.
    • Y-direction: Subject to acceleration from free fall, $ ext{a}_y = -g$.
Effects of Changing Initial Conditions
  • Changes in velocity vector components depend on the initial velocity values.
  • It's important to analyze how changing the angle or magnitude affects the resulting trajectories.
Analysis Model of Projectile Motion
  • Consists of:
    • Constant velocity in horizontal motion.
    • Constant acceleration in vertical motion due to free fall.
Projectile Motion Vectors
  • The final position can be calculated as the vector sum of:
    • Initial position.
    • Initial velocity position.
    • Position resulting from acceleration.
Projectile Motion Implications
  • The vertical component of the velocity is zero at the highest trajectory point.
  • Acceleration remains uniform throughout the trajectory.

Range and Maximum Height of a Projectile

  • Key characteristics include:
    • Range (R): Horizontal distance traveled by the projectile.
    • Maximum height (h): Highest point reached by the projectile.
Height of a Projectile Equation
  • The maximum height can be derived from initial velocity and is applicable for symmetric motion only.
Range of a Projectile Equation
  • The range can also be expressed based on the initial velocity vector under the assumption of symmetric trajectory.
Range of a Projectile: General Findings
  • Maximum range occurs at an angle of $ heta_i = 45^ ext{o}$.
  • Complementary angles yield the same range; however, maximum heights differ, influencing flight times.

Example Problem: Long Jumper

  • A long jumper leaves the ground at $20^ ext{o}$ at a speed of $11.0 ext{ m/s}$.
    • Horizontal distance jumped: Answer: $7.94 ext{ m}$.
    • Maximum height reached: Answer: $0.722 ext{ m}$.

Problem Solving Hints in Projectile Motion

  • Conceptualize: Mental visualization of projectile trajectory.
  • Categorize: Ensure air resistance is disregarded; define coordinate systems.
  • Analyze:
    • Resolve initial velocity into x and y components.
    • Analyze horizontal motion with constant velocity.
    • Analyze vertical motion via constant acceleration techniques.
  • Finalize: Check consistency of answers with mental models, ensuring results are realistic.
Example Problem: Stone Thrown from Building
  • A stone is thrown at a $30^ ext{o}$ angle with $20 ext{ m/s}$ speed from $45 ext{ m}$ high.
    • Time to ground: Answer: $4.22 ext{s}$.
    • Velocity before impact: Answer: $35.8 ext{ m/s}$.

Non-Symmetric Projectile Motion

  • Follow the general projectile motion rules, considering vertical components asymmetrically as needed.
  • Apply problem-solving processes for non-symmetric height attributes.

Uniform Circular Motion

  • Definition: Motion in a circular path at constant speed.
  • Analysis: Involves constant radial acceleration due to changing direction of velocity.
  • The velocity vector remains tangent to the circular path.
Changing Velocity in Uniform Circular Motion
  • Changes in the velocity vector result from direction alterations, impacting the overall motion.

Centripetal Acceleration

  • This acceleration is always directed towards the center of the circle, maintaining perpendicularity to the path.
  • Magnitude of Centripetal Acceleration ($a_c$): Given by the equation:
    • a_c = rac{v^2}{r} where $v$ is speed, $r$ is the radius of circular motion.
Period of Circular Motion
  • Period (T): Time required for one complete revolution, defined as:
    • T = rac{ ext{Circumference}}{v}
Tangential Acceleration
  • Speed changes may occur alongside circular motion, contributing to tangential acceleration.
Total Acceleration
  • Total acceleration arises from both tangential ($a_t$) and centripetal ($a_r$) acceleration:
    • Total Acceleration Vector: Direction changes based on speed changes.

Relative Velocity

  • Observers moving relative to each other observe different outcomes, yet their observations correlate.
  • A frame of reference describes a Cartesian coordinate system where one observer is stationary at the origin.
Different Measurements Examples
  • Observer A and B: Measure point P at different coordinates due to differing reference frames.
  • A moving observer sees relative velocities differently influenced by shared motion factors.
Notation in Relative Velocity
  • The first subscript denotes the observed entity, while the second signifies the observer.
  • Example: Velocity of A as perceived by observer B.
Relative Velocity Equations
  • Positions in two reference frames relate through their velocities, detailed by Galilean transformation equations:
    • PA=PB+vtP_A = P_B + vt
    • extVelocityofparticlePbyA=extVelocityofparticlePbyB+extVelocityofframeAext{Velocity of particle P by A} = ext{Velocity of particle P by B} + ext{Velocity of frame A}

Acceleration in Different Frames of Reference

  • The acceleration of particles is invariant across observers moving at constant velocities relative to each other.