Motion in Two and Three Dimensions Study Notes

INTRODUCTION

• Chapter 4: Motion in Two and Three Dimensions

  • Sections Covered:

  • 4.1 Displacement and Velocity Vectors

  • 4.2 Acceleration Vector

  • 4.3 Projectile Motion

  • 4.4 Uniform and Nonuniform Circular Motion

  • 4.5 Relative Motion in One and Two Dimensions

• Overview:

  • Kinematics extends to describe motion in two and three dimensions. Curved paths dominate natural motion, e.g., footballs, birds, celestial bodies.

  • The Red Arrows, aerobatic jets, exemplify the need for three-dimensional motion analysis. Each jet has unique velocity and position vectors.

  • This chapter begins by elaborating on vectors and how the principles of motion from one dimension can be extrapolated to multi-dimensional contexts.

4.1 Displacement and Velocity Vectors

LEARNING OBJECTIVES

• Calculate position vectors in multi-dimensional problems.

• Solve for displacement in two or three dimensions.

• Calculate the velocity vector given a position vector as a function of time.

• Compute average velocity in multiple dimensions.

Vectors in Motion

• Displacement Vector: The displacement vector in three dimensions is defined concerning a coordinate system. Typically, points are described using coordinates (x, y, z).

  • A position vector from the origin to point P is noted as r such that r = x(â) + y(b̂) + z(ĉ), where â, b̂, ĉ are unit vectors.

  • The right-handed coordinate system is utilized throughout analysis. (Refer to Figures 4.2 and 4.3).

  • The displacement vector Δr from initial position r1 to final position r2 is defined as:
    $ext{Δr} = r<em>2 - r</em>1$

Example 4.1: Polar Orbiting Satellite

• Problem: Calculate the displacement vector from the North Pole to an arbitrary latitude for a satellite.

  • Assumptions:

  • Radius of Earth = 6370 km

  • Altitude of the satellite = 400 km

  • Position vectors can be evaluated using unit vectors based on latitude/longitude.

Example 4.2: Brownian Motion

• Question: Given multiple independent displacement vectors, how to compute total displacement?

  • Total displacement needs vector addition, with each step recorded numerically to form a final result.

Velocity Vector

• The instantaneous velocity vector v is defined as:
  $ext{v} = rac{d ext{r}}{dt}$; In component form, it can be decomposed as:
  $ext{v} = v<em>x(â) + v</em>y(b̂) + v_z(ĉ)$.

Example 4.3: Calculating the Velocity Vector

• Given a particle's position function, compute:

  • (a) Instantaneous velocity at t = 2.0s.

  • (b) Average velocity between t = 1s to 3s.

  • Uses derivative and average definitions:
    $ext{v}_{avg} = rac{ ext{Δr}}{ ext{Δt}}$.

Independence of Perpendicular Motions

• Motion in different axes (x, y, z) is independent, allowing analysis of movement in each direction separately.

• Example: Two baseballs dropped; one horizontal, one vertical. Both share vertical displacement influenced only by gravity, not the horizontal motion.

4.2 Acceleration Vector

LEARNING OBJECTIVES

• Determine the acceleration vector from the velocity function in unit vector notation.

• Describe particle motion with constant acceleration in three dimensions.

• Use one-dimensional motion equations along perpendicular axes for multi-dimensional motion.

Instantaneous Acceleration

• The instantaneous acceleration is derived from: $ext{a} = rac{d ext{v}}{dt}$.

  • Expressed in component forms yields:
    $ext{a} = a<em>x(â) + a</em>y(b̂) + a_z(ĉ)$.

Example 4.4: Finding an Acceleration Vector

• Given particle velocity, determine:

  • (a) Acceleration function by deriving velocity with respect to time.

  • (b) Evaluate at specific time.

Constant Acceleration

• Multidimensional motion still allows usage of standard one-dimensional kinematic equations, adapted for applicable dimensions (x, y, z).

Example 4.6: Skier Down a Slope

• Analyze skier's position and velocity as functions of time. Determine components under stipulated initial parameters.

4.3 Projectile Motion

LEARNING OBJECTIVES

• Use one-dimensional motion in perpendicular directions for projectile analysis.

• Calculate range, time of flight, and maximum height for projectiles.

• Derive projectile trajectory equations.

Principles of Projectile Motion

• Defined under gravitational influence with negligible air resistance, breaking down motion into horizontal (x-axis) and vertical (y-axis) components.

Horizontal Motion Capitalization

• Since gravity affects only the vertical dimension, horizontal motion remains at constant velocity.

Problem-Solving Strategy

1. Resolve projectile motion into horizontal and vertical components.

2. Treat these motions as independent (timed together).

3. Combine findings to compute total displacement and velocity vectors upon trajectory analysis.

Example 4.7: Fireworks Shell Trajectory

• Analyze factors affecting the maximum altitude, time of explosion, horizontal displacement during burst.

4.4 Uniform and Nonuniform Circular Motion

LEARNING OBJECTIVES

• Derive centripetal acceleration for circular motion.

• Use circular motion equations for position, velocity, and acceleration.

• Differentiate between centripetal and tangential acceleration.

Uniform Circular Motion

• Even if speed remains constant, the object changes direction resulting in non-zero centripetal acceleration:
  $a_c = rac{v^2}{r}$ where v = tangential speed and r = radius of circular path.

Example 4.10: Vector Calculation of Centripetal Acceleration

• Calculate radius inducing 1g acceleration on pilot inside a jet executing circular movements.

Nonuniform Circular Motion

• Involves both centripetal and tangential acceleration (change in speed):

  • Total acceleration is the vector sum.

  • Equations derived for both types of acceleration.

Example 4.12: Velocities During Circular Motion

• Find total acceleration with combined centripetal and tangential factors.

4.5 Relative Motion in One and Two Dimensions

LEARNING OBJECTIVES

• Define concepts of reference frames.

• Establish and manipulate position and velocity vector equations for relative motion.

• Graph relative motion scenarios.

Concepts of Reference Frames

• Motion described concerning a specified reference frame (e.g., ground, moving vehicles).

Relative Motion in One Dimension

• Use vector notation and add/subtract velocities depending on frames configured.

• Practical example of calculating velocities in practical scenarios, such as passenger speeds in vehicles.

Example 4.13: Car and Truck Relative Velocity

• Analyze the effect of two vehicles moving towards a shared intersection from different directions using reference frame calculations.

Relative Motion in Two Dimensions

• Enhance concepts into two dimensions:

  • Establish position through combined vectors with respect to targets.

  • Confirm velocity correlations and leverage independence of dimensional movements.

Example 4.14: Flying a Plane in Wind

• Compensate for an external variable affecting direct paths to determine requisite adjustments in velocity.

Chapter Review

Key Terms and Equations

• Key Terms:

  • Acceleration Vector, Centripetal Acceleration, Displacement Vector, Projectile Motion, etc.

• Key Equations:

  • Position and Displacement Vector definitions, Kinematic equations for perpendicular axes, etc.

Conceptual Questions

• Prepare for examinations leveraging application-oriented questions for each section covered.

Problems

• Engage with real-life problems applying principles learned to reinforce understanding.

INTRODUCTION- Chapter 4: Motion in Two and Three Dimensions - Sections Covered: - 4.1 Displacement and Velocity Vectors - 4.2 Acceleration Vector - 4.3 Projectile Motion - 4.4 Uniform and Nonuniform Circular Motion - 4.5 Relative Motion in One and Two Dimensions- Overview: - Kinematics extends to describe motion in two and three dimensions. Curved paths dominate natural motion, e.g., footballs, birds, celestial bodies. - The Red Arrows, aerobatic jets, exemplify the need for three-dimensional motion analysis. Each jet has unique velocity and position vectors. - This chapter begins by elaborating on vectors and how the principles of motion from one dimension can be extrapolated to multi-dimensional contexts.---## 4.1 Displacement and Velocity Vectors### LEARNING OBJECTIVES- Calculate position vectors in multi-dimensional problems.- Solve for displacement in two or three dimensions.- Calculate the velocity vector given a position vector as a function of time.- Compute average velocity in multiple dimensions.### Vectors in Motion- Displacement Vector: The displacement vector in three dimensions is defined concerning a coordinate system. Typically, points are described using coordinates (x, y, z).- A position vector from the origin to point P is noted as r such that $<br> \vec{r} = x\hat{a} + y\hat{b} + z\hat{c}$, where $<br> \hat{a}, \hat{b}, \hat{c}$ are unit vectors. - The right-handed coordinate system is utilized throughout analysis. (Refer to Figures 4.2 and 4.3). - The displacement vector $<br> \Delta\vec{r}$ from initial position $<br> \vec{r}<em>1$ to final position $\vec{r}</em>2$ is defined as:
$<br> \Delta\vec{r} = \vec{r}<em>2 - \vec{r}</em>1$#### Example 4.1: Polar Orbiting Satellite- Problem: Calculate the displacement vector from the North Pole to an arbitrary latitude for a satellite.- Assumptions: - Radius of Earth = 6370 km - Altitude of the satellite = 400 km - Position vectors can be evaluated using unit vectors based on latitude/longitude.#### Example 4.2: Brownian Motion- Question: Given multiple independent displacement vectors, how to compute total displacement?- Total displacement needs vector addition, with each step recorded numerically to form a final result.#### Velocity Vector- The instantaneous velocity vector $<br> \vec{v}$ is defined as:
$<br> \vec{v} = \frac{d\vec{r}}{dt}$; In component form, it can be decomposed as:
$<br> \vec{v} = v<em>x\hat{a} + v</em>y\hat{b} + v<em>z\hat{c}$.#### Example 4.3: Calculating the Velocity Vector- Given a particle's position function, compute:- (a) Instantaneous velocity at $t = 2.0s$. - (b) Average velocity between $t = 1s$ to $3s$. - Uses derivative and average definitions:
$\vec{v}</em>{avg} = \frac{\Delta\vec{r}}{\Delta t}$.### Independence of Perpendicular Motions- Motion in different axes (x, y, z) is independent, allowing analysis of movement in each direction separately.- Example: Two baseballs dropped; one horizontal, one vertical. Both share vertical displacement influenced only by gravity, not the horizontal motion.---## 4.2 Acceleration Vector### LEARNING OBJECTIVES- Determine the acceleration vector from the velocity function in unit vector notation.- Describe particle motion with constant acceleration in three dimensions.- Use one-dimensional motion equations along perpendicular axes for multi-dimensional motion.### Instantaneous Acceleration- The instantaneous acceleration is derived from: $<br> \vec{a} = \frac{d\vec{v}}{dt}$.- Expressed in component forms yields:
$<br> \vec{a} = a<em>x\hat{a} + a</em>y\hat{b} + a<em>z\hat{c}$.#### Example 4.4: Finding an Acceleration Vector- Given particle velocity, determine:- (a) Acceleration function by deriving velocity with respect to time. - (b) Evaluate at specific time.#### Constant Acceleration- Multidimensional motion still allows usage of standard one-dimensional kinematic equations, adapted for applicable dimensions (x, y, z).#### Example 4.6: Skier Down a Slope- Analyze skier's position and velocity as functions of time. Determine components under stipulated initial parameters.---## 4.3 Projectile Motion### LEARNING OBJECTIVES- Use one-dimensional motion in perpendicular directions for projectile analysis.- Calculate range, time of flight, and maximum height for projectiles.- Derive projectile trajectory equations.### Principles of Projectile Motion- Defined under gravitational influence with negligible air resistance, breaking down motion into horizontal (x-axis) and vertical (y-axis) components.#### Horizontal Motion Capitalization- Since gravity affects only the vertical dimension, horizontal motion remains at constant velocity.### Problem-Solving Strategy1. Resolve projectile motion into horizontal and vertical components.2. Treat these motions as independent (timed together).3. Combine findings to compute total displacement and velocity vectors upon trajectory analysis.#### Example 4.7: Fireworks Shell Trajectory- Analyze factors affecting the maximum altitude, time of explosion, horizontal displacement during burst.---## 4.4 Uniform and Nonuniform Circular Motion### LEARNING OBJECTIVES- Derive centripetal acceleration for circular motion.- Use circular motion equations for position, velocity, and acceleration.- Differentiate between centripetal and tangential acceleration.### Uniform Circular Motion- Even if speed remains constant, the object changes direction resulting in non-zero centripetal acceleration:
$a</em>c = \frac{v^2}{r}$ where v = tangential speed and r = radius of circular path.#### Example 4.10: Vector Calculation of Centripetal Acceleration- Calculate radius inducing $1g$ acceleration on pilot inside a jet executing circular movements. ### Nonuniform Circular Motion- Involves both centripetal and tangential acceleration (change in speed):- Total acceleration is the vector sum. - Equations derived for both types of acceleration.#### Example 4.12: Velocities During Circular Motion- Find total acceleration with combined centripetal and tangential factors.---## 4.5 Relative Motion in One and Two Dimensions### LEARNING OBJECTIVES- Define concepts of reference frames.- Establish and manipulate position and velocity vector equations for relative motion.- Graph relative motion scenarios.### Concepts of Reference Frames- Motion described concerning a specified reference frame (e.g., ground, moving vehicles).### Relative Motion in One Dimension- Use vector notation and add/subtract velocities depending on frames configured.- Practical example of calculating velocities in practical scenarios, such as passenger speeds in vehicles.#### Example 4.13: Car and Truck Relative Velocity- Analyze the effect of two vehicles moving towards a shared intersection from different directions using reference frame calculations.#### Relative Motion in Two Dimensions- Enhance concepts into two dimensions:- Establish position through combined vectors with respect to targets. - Confirm velocity correlations and leverage independence of dimensional movements.#### Example 4.14: Flying a Plane in Wind- Compensate for an external variable affecting direct paths to determine requisite adjustments in velocity.---### Chapter Review#### Key Terms and Equations- Key Terms: - Acceleration Vector, Centripetal Acceleration, Displacement Vector, Projectile Motion, etc.- Key Equations: - Position and Displacement Vector definitions, Kinematic equations for perpendicular axes, etc.#### Conceptual Questions- Prepare for examinations leveraging application-oriented questions for each section covered.#### Problems- Engage with real-life problems applying principles learned to reinforce understanding.