Exam 1 Study Guide: Motion in One and Two Dimensions (Chapters 1–3)

Representing Motion

Basics of Motion

• Trajectory: The path that an object follows as it moves through space.

• Motion Diagrams: Visual representations showing an object's position at successive times, useful for analyzing motion qualitatively.

Models and Modelling

• Particle Model: Treats objects as if all their mass is concentrated at a single point, simplifying analysis of motion.

One-Dimensional Models

• Coordinate System: A reference framework (usually a number line) for specifying position.

• Position (x): Location of an object relative to the origin.

• Displacement (\( \Delta x \)): Change in position: $\Delta x = x_f - x_i$

• Time Interval (\( \Delta t \)): Difference between two clock readings: $\Delta t = t_f - t_i$

Velocity

• Uniform Motion: Motion at constant velocity (no acceleration).

• Average Velocity (\( v_{avg} \)): $v_{avg} = \frac{\Delta x}{\Delta t}$

• Speed vs. Velocity: Speed is the magnitude of velocity (always positive); velocity includes direction.

• Units: SI unit for velocity is meters per second (m/s).

Scale and Measurement

• Significant Figures: Digits in a measurement that are known with certainty plus one estimated digit.

• Scientific Notation: Expresses numbers as a product of a coefficient and a power of ten (e.g., $3.0 \times 10^2$).

• Units: Standard units (SI) must be used in calculations.

• Order of Magnitude: Estimating values to the nearest power of ten for quick calculations.

Vectors

• Vector vs. Scalar: Vectors have both magnitude and direction (e.g., velocity, displacement); scalars have only magnitude (e.g., speed, distance).

• Graphical Representation: Vectors are drawn as arrows; length represents magnitude, arrowhead shows direction.

• Displacement Vector: Represents change in position.

• Vector Addition: Combine vectors tip-to-tail or using components.

• Velocity Vectors: Show both speed and direction of motion.

Motion in One Dimension

Describing Motion (Kinematics)

• Motion Diagrams vs. Position-Time Graphs: Diagrams show discrete positions; graphs plot position as a function of time.

• Sign Conventions: Positive and negative directions must be defined (e.g., right is positive).

• Velocity Representation: Slope of position-time graph gives velocity.

• Finding Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Finding Position: $x_f = x_i + v_{avg} \Delta t$ (for constant velocity)

Uniform Motion

• Position Equation: $x = x_0 + vt$

• Proportional Relationships: Displacement is proportional to time for uniform motion.

• Area Under Velocity-Time Graph: Represents displacement.

Instantaneous Velocity

• Definition: Velocity at a specific instant; mathematically, the slope of the tangent to the position-time curve.

• Positive vs. Negative Slope: Positive slope = motion in positive direction; negative slope = motion in negative direction.

Acceleration

• Definition: Rate of change of velocity.

• Equation: $a_{avg} = \frac{\Delta v}{\Delta t}$

• Sign of Acceleration: Positive acceleration increases velocity in the positive direction; negative acceleration (deceleration) decreases it.

• Slowing Down vs. Speeding Up: If velocity and acceleration have opposite signs, the object slows down.

Motion with Constant Acceleration

• Kinematic Equations (The "Big Three"):

  • $v = v_0 + at$

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Quadratic Relationships: Displacement is proportional to the square of time when acceleration is constant.

Free Fall

• Definition: Motion under the influence of gravity only (neglecting air resistance).

• Acceleration Due to Gravity: $g = 9.80\ \mathrm{m/s^2}$ (downward)

• Free Fall Equations: Use kinematic equations with $a = -g$ (if upward is positive).

• Motion Diagrams and Velocity Graphs: Show increasing or decreasing velocity depending on direction of motion.

Vectors and Motion in Two Dimensions

Motion Diagrams in Two Dimensions

• Displacement and Velocity Vectors: Can be determined from successive positions in a motion diagram.

Vector Math

• Addition and Subtraction: Combine vectors graphically (tip-to-tail) or algebraically using components.

• Multiplication by Scalar: Changes magnitude but not direction (unless scalar is negative).

Coordinate Systems in Two Dimensions

• Axes: Typically x (horizontal) and y (vertical).

• Position Vector: $\vec{r} = x\hat{i} + y\hat{j}$

Vector Components

• Finding Components: For a vector of magnitude $A$ at angle $\theta$:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

• Finding Magnitude and Angle:

  • $A = \sqrt{A_x^2 + A_y^2}$

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

Adding/Subtracting Vectors Using Components

• Add or subtract corresponding components: $\vec{C} = \vec{A} + \vec{B} \implies C_x = A_x + B_x$, $C_y = A_y + B_y$

Constant Velocity Motion in Two Dimensions

• On a Ramp: Velocity components depend on the angle of incline.

• Component Equations: $v_x = v \cos \theta$, $v_y = v \sin \theta$

Accelerated Motion on a Ramp

• Gravity's Effect: Acceleration along the ramp is $a = g \sin \theta$

• Angle of Incline: Determines the magnitude of acceleration and velocity components.

Motion in Two Dimensions

• Velocity via Displacement: $\vec{v} = \frac{\Delta \vec{r}}{\Delta t}$

• Acceleration via Change in Velocity: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

• Motion Diagrams: Show both velocity and acceleration vectors at different points.

Projectile Motion

• Horizontal and Vertical Components: Analyze motion separately in x and y directions.

• Kinematic Equations for Each Component:

  • Horizontal: $x = x_0 + v_{0x} t$ (no horizontal acceleration)

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

  • Vertical velocity: $v_y = v_{0y} - g t$

• Analyzing Motion: Use motion diagrams and graphs to visualize projectile trajectories.

Problem Solving in Projectile Motion

• Break initial velocity into components using launch angle.

• Apply kinematic equations to each direction independently.

• Use time of flight, maximum height, and range as key quantities.

Range of a Projectile

• Range Equation (for launch and landing at same height):

  • $R = \frac{v_0^2 \sin 2\theta}{g}$

• Finding Initial Velocity Components:

  • $v_{0x} = v_0 \cos \theta$

  • $v_{0y} = v_0 \sin \theta$

Example: Projectile Launched at Angle

• A ball is launched with speed $v_0$ at angle $\theta$ above the horizontal. Find the time of flight, maximum height, and range.

• Time of Flight: $t = \frac{2 v_0 \sin \theta}{g}$

• Maximum Height: $h_{max} = \frac{v_0^2 \sin^2 \theta}{2g}$

• Range: $R = \frac{v_0^2 \sin 2\theta}{g}$

Additional info: This guide synthesizes the main concepts, equations, and problem-solving strategies for Chapters 1–3, focusing on motion in one and two dimensions, vectors, and projectile motion. Students should practice applying these principles to a variety of problems, including those involving ramps, free fall, and projectiles.