1-D Kinematics and Descriptive Motion Graphing

Fundamental Principles of Position vs. Time Graphs

• The slope of the line on a position vs. time graph represents an object's velocity ($v$). These graphs plot position ($x$) on the vertical y-axis and time ($t$) on the horizontal x-axis.

• Slope Magnitude and Direction:

  • The magnitude (numerical value) of the slope is equal to the object's speed ($s$).

  • The direction of the slope, whether it is upward (positive slope, $+$) or downward (negative slope, $-$), indicates the direction of the velocity vector.

• Visual Representations of Motion on Position-Time Graphs:

  • A horizontal line: Represents an object at rest; there is no movement occurring ($v = 0\,m/s$).

  • A curved line: Represents acceleration ($a \neq 0\,m/s^2$). The slope is changing, therefore the velocity is changing.

  • A straight diagonal line: Represents constant velocity ($a = 0\,m/s^2$). The slope remains the same throughout the motion.

  • A gradually sloped line: Indicates a slow velocity.

  • A steeply sloped line: Indicates a fast (high) velocity.

Multi-Object Analysis of Position vs. Time Graphs

Based on a graph depicting objects A, B, C, D, and E (where A and E are horizontal, B and D are straight diagonal lines with different slopes, and C is a curved line):

• Objects at Rest: Objects A and E are at rest because they are represented by horizontal lines with a slope of zero.

• Objects Not Moving: Mirrors the objects at rest; A and E are not moving.

• Objects Accelerating: Object C is accelerating because it is represented by a curved line.

• Changing Direction: An object changes direction on a position-time graph if the slope changes from positive to negative or negative to positive.

• Object Speed Comparison:

  • Traveling Fastest: The object with the steepest slope is traveling the fastest (Object B in the provided visualization).

  • Traveling Slowest: Among moving objects, the one with the shallowest (most gradual) slope is traveling the slowest (Object C in the provided visualization).

• Direction of Motion: Objects move in the same direction if their slopes have the same sign (positive or negative). Objects moving in the same direction as object B have a similar upward/positive slope.

Fundamental Principles of Velocity vs. Time Graphs

• The slope of the line on a velocity vs. time graph represents an object's acceleration ($a$). These graphs plot velocity ($v$) on the vertical y-axis and time ($t$) on the horizontal x-axis.

• Area Under the Curve: The area bounded by the line and the time axis is equal to the object's displacement ($\Delta x$).

• Visual Representations of Motion on Velocity-Time Graphs:

  • A horizontal line: Represents constant velocity ($a = 0\,m/s^2$). The velocity value is not changing over time.

  • A straight diagonal line: Represents a constant rate of acceleration; the object is either speeding up or slowing down.

  • A gradually sloped line: Represents a small or low acceleration.

  • A steeply sloped line: Represents a large or high acceleration.

Multi-Object Analysis of Velocity vs. Time Graphs

Based on a graph depicting objects A, B, C, D, and E (where A is horizontal, B is a steep positive diagonal, C is a gradual positive diagonal, D is at zero, and E is a negative diagonal):

• Objects at Rest: An object is at rest if its velocity is consistently $0\,m/s$ (Object D).

• Objects Accelerating: Objects B, C, and E are accelerating because they have non-zero slopes.

• Changing Direction: An object changes direction if the line crosses the horizontal time axis (moving from a positive velocity to a negative velocity or vice versa).

• Acceleration Comparison:

  • Greatest Acceleration: Object B has the greatest acceleration because it has the steepest slope.

  • Smallest Acceleration: Among accelerating objects, Object C has the smallest acceleration due to its shallow/gradual slope.

• Direction of Motion: Objects move in the same direction if they are located on the same side of the velocity axis (e.g., both having positive or both having negative velocity values).

Ford F-150 Case Studies: Velocity-Time Graph Data

Analysis of specific motion scenarios for a Ford F-150 vehicle:

• Scenario 1 (Graph #10):

  • Velocity ($v$): $-5\,m/s$ (constant).

  • Speed ($s$): $5\,m/s$.

  • Displacement ($\Delta x$): $-90\,m$.

  • Distance ($d$): Although the label lists $0\,m$, standard calculation for distance traveled at $5\,m/s$ for the duration required for $-90\,m$ displacement ($18\,s$) would be $90\,m$.

• Scenario 2 (Graph #11):

  • Velocity ($v$): $-10\,m/s$ (constant).

  • Speed ($s$): $10\,m/s$.

  • Displacement ($\Delta x$): $-50\,m$ (Calculated as $\text{Area} = -10\,m/s \times 5\,s$).

  • Distance ($d$): $50\,m$.

• Scenario 3 (Graph #12):

  • Velocity ($v$): $-5\,m/s$ (constant).

  • Speed ($s$): $5\,m/s$.

  • Displacement ($\Delta x$): $-25\,m$.

  • Distance ($d$): $25\,m$.

Matching Verbal Descriptions to Kinematic Graphs

• Description A: The object is moving fast with a constant velocity and then moves slow with a constant velocity.

  • Position-Time Graph Characteristics: A steep straight line followed by a more gradual straight line.

• Description B: The object is moving in one direction with a constant rate of acceleration (slowing down), changes directions, and continues in the opposite direction with a constant rate of acceleration (speeding up).

  • Velocity-Time Graph Characteristics: A straight diagonal line crossing the horizontal axis.

• Description C: The object moves with a constant velocity and then slows down.

  • Velocity-Time Graph Characteristics: A horizontal line followed by a diagonal line sloping toward the zero-velocity axis.

• Description D: The object moves with a constant velocity and then speeds up.

  • Velocity-Time Graph Characteristics: A horizontal line followed by a diagonal line sloping away from the zero-velocity axis.

• Description E: The object maintains a rest position for several seconds and then accelerates.

  • Velocity-Time Graph Characteristics: A line at $v = 0$ for an interval, followed by a sloped line.

Dot Diagrams and Ticker Tapes

Dot diagrams (or ticker tapes) are used to represent the position of an object at equal time intervals.

• Constant Velocity: Dots are spaced evenly. If the velocity is fast, the dots are spaced far apart. If the velocity is slow, the dots are close together.

• Acceleration (Speeding Up): The spacing between consecutive dots increases over time.

• Deceleration (Slowing Down): The spacing between consecutive dots decreases over time.

• Directionality: An arrow must be placed on the ticker tape to indicate the direction of motion (left or right).

Definitions

• Distance: Total path traveled, always positive.

• Displacement: Change in position that includes direction; can be positive, negative, or zero (e.g., running a full lap and finishing where started results in distance ≠ 0 but displacement = 0).

• Speed: The rate at which distance is covered; calculated as speed = distance ÷ time.

• Velocity: Speed with direction; indicates how fast something moves and in what direction.

• Acceleration: The rate of change of velocity; can indicate speeding up, slowing down, or changing direction.

• Constant Velocity: A scenario where an object travels at the same speed and direction over time.

• Direction of Motion: The specific path that an object travels relative to a reference point, such as left, right, forward, or backward.

Position-Time (x-t) Graphs

• Helps determine if an object is moving or stopped.

• Indicates if moving toward or away from the origin based on slope.

• Compares speeds by examining slopes:
  
  

  • Positive slope: Motion in the positive direction.

  • Negative slope: Motion in the negative direction.

  • Horizontal line: Object is not moving.

  • Steeper slope: Represents faster speed.

Velocity-Time (v-t) Graphs

• Matches motion descriptions to graphical representations.

• Determines when an object is moving left or right based on velocity signs.

• Assesses if an object is speeding up or slowing down via slope changes.

• Area under the graph represents displacement:
  
  

  • Positive velocity: Moving right/forward.

  • Negative velocity: Moving left/backward.

  • Horizontal line: Indicates constant velocity.

Interpreting Motion

• Identify motion characteristics such as:
  
  

  • Speeding up

  • Slowing down

  • Constant speed

  • Standing still

  • Changing direction.

• Describe motions accurately based on graphical data.

Metric System and Prefixes

• Common prefixes include:
  
  

  • kilo (k): 1000

  • hecto (h): 100

  • deca (da): 10

  • centi (c): 0.01

  • milli (m): 0.001

• Examples:
  
  

  • 1 km = 1000 m

  • 1 m = 1000 mm

  • 1000 g = 1 kg

Unit Conversions

• Be able to convert between various units including:
  
  

  • km ↔ m ↔ mm

  • g ↔ kg

  • gal ↔ L ↔ mL

  • mi/hr ↔ ft/min

  • m/s ↔ km/hr

• Tips for conversion:
  
  

  • Write conversion factors as fractions and cancel units.

  • Verify that only the desired unit remains.

Scientific Notation

• Skills:
  
  

  • Convert standard numbers to scientific notation and vice versa.

  • Compare numbers in scientific notation.

• Rules:
  
  

  • Number must be between 1 and 10.

  • Positive exponent indicates a large number, while negative exponent signifies a small decimal.

Math Skills

• Practice evaluating expressions with substitutions, using order of operations, and working with positive and negative integers.

• Examples include substituting values into formulas and solving expressions with parentheses and exponents.

Distance vs. Displacement

• Key differences to know:
  
  

  • Distance is the total path traveled and is always positive.

  • Displacement signifies change in position and accounts for direction, indicating results as positive, negative, or zero, following the previous example of running a lap.

• Be able to convert between various units including:
  
  

  • km ↔ m ↔ mm
    
    

  • g ↔ kg
    
    

  • gal ↔ L ↔ mL
    
    

  • mi/hr ↔ ft/min
    
    

  • m/s ↔ km/hr
    

• Tips for conversion:
  
  

  • Write conversion factors as fractions and cancel units.
    
    

  • Verify that only the desired unit remains.