Kinematics Unit: 1D Sub-Unit Notes

Kinematics Unit: 1D Sub-Unit

1.1.1 Displacement and Distance
  • Scalar: A measured quantity that has no direction.
    • Examples: Distance, Time, Mass, Volume.
  • Vector: A measured quantity that includes direction.
    • The sign indicates direction.
    • Example: Displacement.
Definitions
  • Distance: The complete length of a path traveled by a moving object.
    • Is a Scalar quantity.
    • Examples:
    • A ball rolls 5 meters north: Distance = 5 m5 \text{ m}
    • A cat runs 8 meters west: Distance = 8 m8 \text{ m}
    • A bird flies 5 meters north, then 7 meters south: Distance = 5 m+7 m=12 m5 \text{ m} + 7 \text{ m} = 12 \text{ m}
  • Displacement: The length of the straight-line path of a moving object from its origin to its final position.
    • Is a Vector quantity.
    • Examples:
    • A ball rolls 5 meters north: Displacement = +5 m+5 \text{ m}
    • A cat runs 8 meters west: Displacement = −8 m-8 \text{ m}
    • A bird flies 5 meters north, then 7 meters south: Displacement = (+5 m)+(−7 m)=−2 m(+5 \text{ m}) + (-7 \text{ m}) = -2 \text{ m}
Sign Conventions
  • Positive Direction:
    • Up
    • North
    • East
    • Right
  • Negative Direction:
    • Down
    • South
    • West
    • Left
Examples: Distance and Displacement Calculation
  • Example #1: A man drives his car 3 miles north, then 4 miles east.
    • Distance traveled: 3 miles+4 miles=7 Miles3 \text{ miles} + 4 \text{ miles} = 7 \text{ Miles}
    • Displacement from his point of origin: Using the Pythagorean theorem for perpendicular movements: (3 miles)2+(4 miles)2=9+16=25=5 Miles\sqrt{(3 \text{ miles})^2 + (4 \text{ miles})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ Miles} in the Northeast direction.
  • Example #2: Three men leave the same house on foot.
    • Man #1: Walks 30 feet north, then 40 feet west.
    • Distance = 30 ft+40 ft=70 ft30 \text{ ft} + 40 \text{ ft} = 70 \text{ ft}
    • Displacement = (30 ft)2+(40 ft)2=900+1600=2500=50 ft\sqrt{(30 \text{ ft})^2 + (40 \text{ ft})^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ ft} (NW directon)
    • Man #2: Walks 90 feet south, then 88 feet north.
    • Distance = 90 ft+88 ft=178 ft90 \text{ ft} + 88 \text{ ft} = 178 \text{ ft}
    • Displacement = (−90 ft)+(+88 ft)=−2 ft(-90 \text{ ft}) + (+88 \text{ ft}) = -2 \text{ ft} (2 ft South)
    • Man #3: Walks 10 feet east, then 50 feet west.
    • Distance = 10 ft+50 ft=60 ft10 \text{ ft} + 50 \text{ ft} = 60 \text{ ft}
    • Displacement = (+10 ft)+(−50 ft)=−40 ft(+10 \text{ ft}) + (-50 \text{ ft}) = -40 \text{ ft} (40 ft West)
    • Which man has traveled the greatest distance? Man #2 (178 ft178 \text{ ft}).
    • Who is farthest from the house? Man #1 (50 ft50 \text{ ft}).
    • Who is closest to the house? Man #2 (2 ft2 \text{ ft}).
Distance vs. Time Graph
  • Slope: Represents speed.
  • Characteristics: Distance cannot be negative.
  • Interpretation:
    • If the line is horizontal, the object is not moving (speed is zero).
    • If the line is straight and sloping upwards, the object is moving at a constant positive speed.
    • If the line is curved and sloping upwards, the object's speed is changing (accelerating or decelerating).