Describing Motion Study Guide

Chapter Vocabulary & Core Concepts

• Lesson 1 Vocabulary:

  • Reference Point: A starting point used to describe the position or location of an object.

  • Position: An object's distance and direction from a reference point.

  • Motion: The process of changing position relative to a reference point.

  • Displacement: The difference between the initial position and the final position of an object.

  • Specify (Academic Vocabulary): To indicate or describe in detail.

  • Dimension (Academic Vocabulary): Distance or length measured in one direction.

• Lesson 2 Vocabulary:

  • Speed: A measure of the distance an object travels per unit of time.

  • Constant Speed: The rate of change of position in which the same distance is traveled each second.

  • Instantaneous Speed: Speed at a specific instant in time.

  • Average Speed: The total distance traveled divided by the total time taken to travel that distance.

  • Velocity: The speed and direction of a moving object.

• Lesson 3 Vocabulary:

  • Acceleration: A measure of the change in velocity during a period of time.

  • Average Acceleration: A change in velocity during a time interval divided by the time interval during which the velocity changes.

Describing Position and Reference Points

• Use of Reference Points:

  • A reference point combined with a direction and a distance defines an object's position.

  • Reference Point + Direction + Distance $\rightarrow$ Position.

• Changing Reference Points:

  • When a reference point is changed, the description of both the direction and the distance to the object must also be changed.

• One-Dimensional Position Relative to Axis (Specified Direction: West):

  • Along a linear path containing a Mall, Coffee Shop, Library, and Post Office:

  • Mall to Post Office: Reference Direction = negative, Distance = $60\,\text{km}$.

  • Library to Coffee Shop: Reference Direction = positive, Distance = $10\,\text{km}$.

  • Mall to Library: Reference Direction = negative, Distance = $40\,\text{km}$.

  • Post Office to Coffee Shop: Reference Direction = positive, Distance = $30\,\text{km}$.

  • Directions from Post Office to Mall: Travel west $60\,\text{km}$ past the library and/or the coffee shop.

Describing Position in Two Dimensions & Changes in Position

• Sequence for Describing Position in Two Dimensions:

  1. Choose a reference point.

  2. Specify two reference directions.

  3. Determine distance from the reference point along each reference direction.

• Determining Motion:

  • Motion is defined as the process of changing position.

  • To determine whether an object is in motion, compare the object's position with that of a reference point.

  • If the distance between the object and the reference point changes, motion has occurred.

• Distinguishing Distance and Displacement:

  • Distance: The total length of a path an object follows.

  • Displacement: The difference between the initial position and the final position of an object.

• Motion Analysis Scenario (Riding on a Wagon):

  • If the wagon is used as the reference point, a passenger riding on the wagon is not moving because their position relative to the wagon remains constant.

  • The wagon is known to be in motion relative to the environment because the distance between the wagon and other environmental objects changes.

Fundamentals of Speed and Velocity

• Calculating Speed:

  • Speed is defined as a measure of the distance an object travels per unit of time.

  • Speed is determined by dividing the distance traveled by the time it takes to travel that distance.

• Typical Measurements of Speed:

  • Person walking: $6.4\,\text{km/h}$

  • Car on highway: $60\text{--}100\,\text{km/h}$

  • Airplane flying: $245\,\text{m/s}$

• Types of Speed:

  • Constant Speed: The rate of change of position in which the same distance is traveled each second.

  • Changing Speed: Motion in which the distance traveled is different each second.

  • Instantaneous Speed: The speed of an object at a specific instant in time.

• Average Speed Equation:

  • $\text{average speed (in m/s)} = \frac{\text{total distance (in m)}}{\text{total time (in s)}}$

  • Symbolic equation: $v = \frac{d}{t}$

• Worked Example (Scenario Calculation):

  • An individual drives $50\,\text{km/h}$ for one hour and $100\,\text{km/h}$ for one hour.

  • Instantaneous speed each hour: $50\,\text{km/h}$ for the first hour and $100\,\text{km/h}$ for the second hour.

  • Average speed for the two hours: $75\,\text{km/h}$.

Analyzing Distance-Time Graphs

• Interpreting Distance-Time Graphs (Race to the Swing):

  • When comparing two runners (a boy and a girl) starting from the school door, when the girl reaches the swing at $30\,\text{m}$, the boy is positioned $20\,\text{m}$ from the school door.

• Sequence of 5 Steps to Calculate Average Speed on a Distance-Time Graph:

  1. Choose one point on the line.

  2. Choose a second point on the line.

  3. Find the time difference of the points.

  4. Find the distance difference of the points.

  5. Divide difference in distance by difference in time.

• Worked Example (Runner's Speed between 4 s and 6 s):

  • Step 1: First point $(4, 60)$

  • Step 2: Second point $(6, 80)$

  • Step 3: Difference in time: $6\,\text{s} - 4\,\text{s} = 2\,\text{s}$

  • Step 4: Difference in distance: $80\,\text{m} - 60\,\text{m} = 20\,\text{m}$

  • Step 5: Calculation: $\frac{20\,\text{m}}{2\,\text{s}} = 10\,\text{m/s}$

  • Runner's Average Speed: $10\,\text{m/s}$

• Motions Illustrated on Distance-Time Graphs:

  • Distance-time graphs can illustrate three specific changes in motion: slowing down, stopping, and speeding up.

• Worked Example (Marble Roll Graph):

  • Starting point: $(0, 0)$

  • Ending point: $(10, 40)$

  • Change in distance: $40\,\text{m}$

  • Change in time: $10\,\text{s}$

  • Average speed calculation: $40\,\text{m} \div 10\,\text{s} = 4\,\text{m/s}$

• Describing Motion with Distance-Time Graphs:

  • A distance-time graph shows how the speed of an object changes over time.

  • It graphically indicates whether an object is speeding up, slowing down, or remaining stopped.

Velocity and Direction of Motion

• Definition of Velocity:

  • Velocity is defined as the speed and direction of motion.

• Conditions for Velocity Change:

  • Velocity changes under any of three conditions:

  1. Direction changes.

  2. Speed changes.

  3. Both speed and direction change.

Understanding Acceleration

• Definition of Acceleration:

  • Acceleration is defined as a measure of the change in velocity during a period of time.

• Three Ways an Object Can Accelerate:

  1. Slow down

  2. Speed up

  3. Change direction

• Vehicle Acceleration Dynamics at Intersections:

  • Leaving an intersection: Acceleration increases in the same direction as that in which the car is moving.

  • Approaching an intersection: Acceleration increases in the opposite direction from that in which the car is moving.

• Vector Direction Relationships (Velocity vs. Acceleration):

  • When an object speeds up, its velocity vector and acceleration vector point in the same direction.

  • When an object slows down, its acceleration vector points opposite to its velocity vector.

  • When an object turns, its acceleration vector points in a direction angled toward the inside of the turn.

Calculating Acceleration and Directional Motion

• Average Acceleration Definition and Formula:

  • Average acceleration is the change in velocity during a time interval divided by the time interval during which the velocity changes.

  • Equation: $a = \frac{v_f - v_i}{t}$

• Variables in the Average Acceleration Equation:

  • $a$ = average acceleration

  • $v_i$ = initial speed

  • $v_f$ = final speed

  • $t$ = time

• Worked Acceleration Calculation (Rocket Scenario):

  • A rocket accelerates from $0$ to $20\,\text{km/s}$. Five seconds after reaching $20\,\text{km/s}$, the rocket is traveling at $280\,\text{km/s}$.

  • Initial speed ($v_i$): $20\,\text{km/s}$

  • Final speed ($v_f$): $280\,\text{km/s}$

  • Total time ($t$): $5\,\text{seconds}$

  • Formula evaluation: $a = \frac{280\,\text{km/s} - 20\,\text{km/s}}{5\,\text{s}} = \frac{260\,\text{km/s}}{5\,\text{s}} = 52\,\text{km/s}$

• Direction of Motion Determination from Acceleration Data:

  • Object 1 (Negative Acceleration Trend):

  • Time $0\,\text{s} \rightarrow 0\,\text{m/s}$, Time $1\,\text{s} \rightarrow -2\,\text{m/s}$, Time $2\,\text{s} \rightarrow -4\,\text{m/s}$, Time $3\,\text{s} \rightarrow -6\,\text{m/s}$, Time $4\,\text{s} \rightarrow -8\,\text{m/s}$.

  • Direction of motion: negative.

  • Object 2 (Positive Acceleration Trend):

  • Time $0\,\text{s} \rightarrow 0\,\text{m/s}$, Time $1\,\text{s} \rightarrow 2\,\text{m/s}$, Time $2\,\text{s} \rightarrow 4\,\text{m/s}$, Time $3\,\text{s} \rightarrow 6\,\text{m/s}$, Time $4\,\text{s} \rightarrow 8\,\text{m/s}$.

  • Direction of motion: positive.

Speed-Time Graphs and Motion Summaries

• Purpose of Speed-Time Graphs:

  • A speed-time graph indicates how speed changes over time.

• Comparative Graph Sets (Distance-Time vs. Speed-Time):

  • SET 1: Distance-time graph shows a horizontal flat line above zero; Speed-time graph shows a line at zero speed. Description: The object is at rest.

  • SET 2: Distance-time graph shows a straight sloped line upward; Speed-time graph shows a horizontal flat line at a constant speed value. Description: The object is moving at a constant speed.

  • SET 3: Distance-time graph shows an upward curved line; Speed-time graph shows a straight sloped line upward from zero. Description: The object is speeding up.

• Car Trip Graph Synthesis (Piecewise Trip Description):

  • Interval 0--20 s: Car starts from rest and accelerates to $35\,\text{km/h}$ in $20\,\text{seconds}$ (positive acceleration).

  • Interval 20--40 s: Car travels at a constant speed of $35\,\text{km/h}$ for $20\,\text{seconds}$ (zero acceleration).

  • Interval 40--50 s: Car slows to a stop in $10\,\text{seconds}$ (negative acceleration).

  • Interval 50--70 s: Car remains at rest for $20\,\text{seconds}$ (zero acceleration).

• Five Primary Ways Motion Can Be Described:

  1. Position

  2. Average speed

  3. Instantaneous speed

  4. Velocity

  5. Acceleration

Motion Application and Graph Analysis

• Describing Motion of a Boat Leaving a Dock:

  • Reference Point: The dock is designated as the reference point.

  • Direction: Away from the dock.

  • Graph Usage: A distance-time graph can be created to track the boat's motion and used to calculate both its average speed and average acceleration over time.