Chapter 3 Reading: 3.1 Position, Displacement, and Average Velocity

Learning Objectives

• Define position, displacement, and distance traveled.

• Calculate total displacement given position as a function of time.

• Determine total distance traveled.

• Calculate average velocity given displacement and elapsed time.

Fundamental Questions of Kinematics

• Describing the motion of an object requires answering three core kinematic questions:

  • Where are you?

  • Where are you going?

  • How fast are you getting there?

• Answering these questions requires specifying position, displacement, and average velocity.

Position and Reference Frames

• Position ($x$): The precise location of an object at any particular time relative to a chosen frame of reference.

• Frame of Reference: An arbitrary set of axes from which the position and motion of an object are described.

  • Stationary Reference Frames: Earth is commonly used as a stationary reference frame, describing positions relative to fixed features. Examples include:

  • Describing a rocket launch relative to Earth as a whole.

  • Describing a cyclist's position relative to stationary buildings passed along a street.

  • Moving Reference Frames: Reference frames can also move relative to Earth. For example, describing the position of a passenger on an airplane utilizes the airplane itself as the reference frame rather than Earth.

• Variables Used for Position:

  • One-dimensional horizontal motion uses the variable $x$.

  • Vertical motion (such as free fall) uses the variable $y$.

Displacement

• Definition: Displacement is the change in position of an object relative to its reference frame. It signifies that an object has moved or been displaced.

• Mathematical Formula: $\Delta x = x_f - x_0$

  • $\Delta x$: Displacement

  • $x_f$: Final position

  • $x_0$: Initial position

  • The uppercase Greek letter delta ($\Delta$) represents "change in" the quantity that follows it.

• Vector Properties:

  • Displacement is a vector quantity, meaning it has both magnitude and direction.

  • Direction is indicated by a positive or negative sign, depending on the chosen positive directional coordinate axis.

  • The SI unit for displacement is the meter ($\text{m}$). Other length units such as kilometers ($\text{km}$) are also used and must be converted to meters for standard calculations.

• Total Displacement:

  • When motion consists of multiple sequential displacements, total displacement ($\Delta x_{\text{Total}}$) is the vector sum of all individual displacements: $\Delta x_{\text{Total}} = \sum \Delta x_i$

  • Pacing Professor Example:

  • A professor moves $2\,\text{m}$ to the right (positive direction) relative to a whiteboard, and then $4\,\text{m}$ to the left.

  • First displacement: $\Delta x_1 = x_1 - x_0 = 2\,\text{m} - 0\,\text{m} = 2\,\text{m}$

  • Second displacement: $\Delta x_2 = x_2 - x_1 = -2\,\text{m} - 2\,\text{m} = -4\,\text{m}$

  • Total displacement: $\Delta x_{\text{Total}} = \Delta x_1 + \Delta x_2 = 2\,\text{m} + (-4\,\text{m}) = -2\,\text{m}$ along the $x$-axis.

• Magnitude of Displacement vs. Distance Traveled:

  • Magnitude of Displacement: The absolute value or size of the displacement vector, which is always non-negative: $\text{Magnitude} = |\Delta x|$

  • In the pacing professor example, individual displacement magnitudes are $|2\,\text{m}| = 2\,\text{m}$ and $|-4\,\text{m}| = 4\,\text{m}$. The magnitude of total displacement is $|-2\,\text{m}| = 2\,\text{m}$.

  • Distance Traveled ($x_{\text{Total}}$): The total length of the path traveled between two positions. It is obtained by summing the magnitudes of all individual displacements: $x_{\text{Total}} = \sum |\Delta x_i|$

  • In the pacing professor example: $x_{\text{Total}} = |\Delta x_1| + |\Delta x_2| = 2\,\text{m} + 4\,\text{m} = 6\,\text{m}$

Average Velocity

• Time Variable ($t$): Introducing time allows kinematics to describe how fast position changes.

  • Each position $x_i$ is assigned a specific time $t_i$.

  • Elapsed Time ($\Delta t$): The time taken to travel between two points: $\Delta t = t_2 - t_1$

• Definition: Average velocity ($\bar{v}$) is a vector quantity defined as the total displacement between two points divided by the elapsed time.

• Mathematical Formula: $\bar{v} = \frac{\text{Displacement between two points}}{\text{Time needed to make the displacement}}$ $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$

• Vector Properties:

  • Average velocity can be positive or negative, depending on the sign of the displacement $\Delta x$.

Example 3.1: Delivering Flyers

• Scenario: Jill sets out from her home ($x_0 = 0\,\text{km}$, $t_0 = 0\,\text{min}$) to deliver yard sale flyers along her street, moving due east (designated as the positive direction).

  • Leg 1: Travels $0.5\,\text{km}$ east in $9\,\text{min}$, running out of flyers ($x_1 = 0.5\,\text{km}$, $t_1 = 9\,\text{min}$).

  • Leg 2: Retraces her steps back to her house ($x_2 = 0\,\text{km}$) taking an additional $9\,\text{min}$ ($t_2 = 18\,\text{min}$).

  • Leg 3: Picks up more flyers and travels east again to $1.0\,\text{km}$ from her house ($x_3 = 1.0\,\text{km}$) taking $15\,\text{min}$ ($t_3 = 33\,\text{min}$).

  • Leg 4: Turns back west toward her house, traveling $1.75\,\text{km}$ west over $25\,\text{min}$ before stopping to rest ($x_4 = -0.75\,\text{km}$, $t_4 = 58\,\text{min}$).

• Tabulated Physical Quantities:   | Time $t_i\,\text{(min)}$ | Position $x_i\,\text{(km)}$ | Displacement $\Delta x_i\,\text{(km)}$ |   |---|---|---|   | $t_0 = 0$ | $x_0 = 0$ | $\Delta x_0 = 0$ |   | $t_1 = 9$ | $x_1 = 0.5$ | $\Delta x_1 = x_1 - x_0 = 0.5$ |   | $t_2 = 18$ | $x_2 = 0$ | $\Delta x_2 = x_2 - x_1 = -0.5$ |   | $t_3 = 33$ | $x_3 = 1.0$ | $\Delta x_3 = x_3 - x_2 = 1.0$ |   | $t_4 = 58$ | $x_4 = -0.75$ | $\Delta x_4 = x_4 - x_3 = -1.75$ |

• Questions & Calculations:

  1. Total Displacement: $\Delta x_{\text{Total}} = \sum \Delta x_i = 0.5\,\text{km} - 0.5\,\text{km} + 1.0\,\text{km} - 1.75\,\text{km} = -0.75\,\text{km}$

  2. Magnitude of Total Displacement: $|\Delta x_{\text{Total}}| = |-0.75\,\text{km}| = 0.75\,\text{km}$

  3. Average Velocity: $\bar{v} = \frac{\text{Total displacement}}{\text{Elapsed time}} = \frac{-0.75\,\text{km}}{58\,\text{min}} = -0.013\,\text{km/min}$

  4. Total Distance Traveled: $x_{\text{Total}} = \sum |\Delta x_i| = 0.5\,\text{km} + 0.5\,\text{km} + 1.0\,\text{km} + 1.75\,\text{km} = 3.75\,\text{km}$

  5. Graphing: Jill's position versus time can be plotted as a visual aid to summarize her motion.

• Significance and Interpretation:

  • A total displacement of $-0.75\,\text{km}$ means Jill ends her trip $0.75\,\text{km}$ due west of her house.

  • An average velocity of $-0.013\,\text{km/min}$ means that if another person walked continuously due west at $0.013\,\text{km/min}$ starting at the same time, both would reach the final stopping point at the exact same moment.

  • If Jill ended her trip back at her house, both her total displacement and average velocity would equal zero.

  • Over the $58\,\text{min}$ elapsed time, her total distance traveled is $3.75\,\text{km}$.

Check Your Understanding 3.1

• Problem Statement:   A cyclist rides $3\,\text{km}$ west and then turns around and rides $2\,\text{km}$ east.

  • (a) What is their displacement?

  • (b) What is the distance traveled?

  • (c) What is the magnitude of their displacement?

• Solution Analysis:

  • Setting East as positive ($+x$) and West as negative ($-x$):

  • First leg displacement: $\Delta x_1 = -3\,\text{km}$

  • Second leg displacement: $\Delta x_2 = +2\,\text{km}$

  • (a) Displacement: $\Delta x = \Delta x_1 + \Delta x_2 = -3\,\text{km} + 2\,\text{km} = -1\,\text{km}$

  • The displacement is $-1\,\text{km}$ (or $1\,\text{km}$ west).

  • (b) Distance Traveled: $x_{\text{Total}} = |\Delta x_1| + |\Delta x_2| = |-3\,\text{km}| + |2\,\text{km}| = 3\,\text{km} + 2\,\text{km} = 5\,\text{km}$

  • (c) Magnitude of Displacement: $|\Delta x| = |-1\,\text{km}| = 1\,\text{km}$