One-Dimensional Motion: Position, Time, and Displacement

Quantitative Foundations of One-Dimensional Motion

• Quantitative measurements are essential for developing a precise, scientific understanding of physical motion.

• Describing motion accurately using a motion diagram requires determining two core quantities:

  • Position: Where an object is located.

  • Time: When the object was at that specific position.

• One-Dimensional ($1\text{-D}$) Motion: Motion of an object that is strictly constrained to move along a single straight line.

• Examples of one-dimensional motion include:

  • A car traveling along a long, straight country road.

  • An airplane taxiing down a straight runway.

  • An elevator moving vertically up and down a shaft.

  • A rock falling vertically downward under the influence of gravity.

  • A skier sliding down a straight mountain slope.

Coordinate Systems and Position Measurement

• Position Definition: An object's explicit location at a specific instant in time.

• Specifying an object's position along a straight line requires three distinct pieces of information:

  • Reference Point (Origin): A fixed reference point from which all measurements and distances originate. The physical placement of the origin is mathematically arbitrary and can be chosen for convenience (such as a post office, a street intersection, or the ground level).

  • Distance: The magnitude of space between the object and the origin, measured along an imaginary axis marked off in equal divisions (such as inches, meters, or miles) with the zero mark located precisely at the origin.

  • Direction / Side of Origin: The side of the origin on which the object is located, specified by extending the axis with increasing positive numbers in one direction and increasing negative numbers in the opposite direction.

• Coordinate System: A structure composed of a fixed origin and an axis marked in both positive and negative directions used to unambiguously define the location of an object.

• Coordinate: A mathematical symbol used to denote a location along an axis.

  • Horizontal motion typically uses the coordinate symbol $x$.

  • Vertical motion typically uses the coordinate symbol $y$.

• Key Applications and Named Examples:

  • Driving $4\,mi$ east of a post office utilizes the post office as the origin ($x = 0$), $4\,mi$ as the spatial distance, and east as the positive direction.

  • A vertical snow depth gauge measures snowfall accumulation using an axis where the road level serves as the natural physical origin.

  • A sprinter running a $50\,m$ race utilizes a horizontal coordinate system marked in meters (abbreviated $m$).

  • Replacing verbose verbal statements (e.g., "the car is at position $4\,mi$") with coordinate symbols (e.g., "the cow is at position $x = 4\,mi$") enables rigorous mathematical analysis of position, velocity, and acceleration.

Time Representation and Clock Origins

• A complete motion diagram represents how an object's position continuously changes as time progresses.

• Frames in a motion diagram must be labeled with precise clock or stopwatch readings (symbol $t$) rather than simple arbitrary sequential frame counts ($0$, $1$, $2$).

• Time Scale Impact on Physical Interpretation:

  • In a motion diagram of a braking car, frame captures recorded $1\,s$ apart depict a slow, leisurely stop.

  • In the same spatial diagram, frame captures recorded $\frac{1}{10}\,s$ apart depict an abrupt, screeching halt.

• Origin of Time ($t = 0\,s$):

  • The choice of $t = 0\,s$ represents the exact instant a stopwatch or clock is started.

  • The placement of $t = 0\,s$ is arbitrary and can be selected at any point in the object's motion depending on the specific phase of interest.

  • A video frame labeled $t = 4\,s$ denotes an event occurring exactly $4\,s$ after the clock was initiated.

  • Setting $t = 0\,s$ typically marks the designated "beginning" of a physics problem, even if the object was moving prior to that moment (such as a car traveling at constant speed before applying brakes).

  • Clock readings and time intervals are abbreviated using $s$ for seconds.

Mathematical Quantification of Displacement

• Subscript Conventions:

  • Initial values at the start of a problem are denoted by the subscript $i$ (e.g., $x_i$ for initial position, $t_i$ for initial time).

  • Final values at the end of a problem are denoted by the subscript $f$ (e.g., $x_f$ for final position, $t_f$ for final time).

  • Multiple distinct intermediate points are designated with numerical subscripts (e.g., $1$, $2$, $3$).

• Displacement ($\Delta x$):

  • Displacement is defined as a change in position over time.

  • The Greek letter delta ($\Delta$) represents a change in a given physical quantity.

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

  • The notation $\Delta x$ represents a single unified symbol; the delta symbol ($\Delta$) cannot be isolated or canceled algebraically.

  • Universal Change Principle: The change in any physical quantity throughout physics is calculated as its final value minus its initial value.

• Signed Nature of Displacement:

  • Displacement is a signed quantity that encapsulates both magnitude and direction.

  • A positive displacement ($\Delta x > 0$) signifies movement toward the positive direction along the axis (e.g., right or east).

  • A negative displacement ($\Delta x < 0$) signifies movement toward the negative direction along the axis (e.g., left or west).

• Detailed Example: Sam's Displacement

  • Spatial Setup: Sam stands $50\,ft$ east of the intersection of 12th Street and Vine. The intersection serves as the origin ($x = 0$).

  • Initial Position: $x_i = +50\,ft$ (positive value indicates east of the origin).

  • Walking East Case: Sam walks to a point $150\,ft$ east of Vine ($x_f = +150\,ft$).     $\Delta x = x_f - x_i = 150\,ft - 50\,ft = 100\,ft$     Sam undergoes a displacement of $100\,ft$ to the east.

  • Walking West Case: Sam starts at $x_i = +50\,ft$ and walks back to the intersection ($x_f = 0\,ft$).     $\Delta x = x_f - x_i = 0\,ft - 50\,ft = -50\,ft$     The negative sign indicates a displacement of $50\,ft$ to the west.

Time Intervals and Temporal Measurement

• Time Interval ($\Delta t$):

  • A time interval measures the total elapsed time as an object moves from an initial position $x_i$ at time $t_i$ to a final position $x_f$ at time $t_f$.

  • Mathematical Formula:     $\Delta t = t_f - t_i$

  • Sign Property: Unlike displacement $\Delta x$, a time interval $\Delta t$ is ALWAYS positive because $t_f$ is always greater than $t_i$.

Worked Example: Emily's Bicycle Ride

• Context: Emily goes on a bicycle ride along a straight east-west country road past a water tower.

• Initial State ($t_i = \text{noon}$): Emily is located $3\,mi$ east of the water tower.

• Final State ($t_f = \text{noon} + 30\,\text{min}$): A half-hour later, Emily is located $2\,mi$ west of the water tower.

• Step-by-Step Problem-Solving Method:

  • Strategize: Model Emily using the particle model along a one-dimensional coordinate system aligned with the straight road.

  • Prepare: Establish an x-axis along the road where increasing $x$ corresponds to moving east. Set the origin $x = 0$ at the water tower. Identify coordinates:     $x_i = +3\,mi$     $x_f = -2\,mi$

  • Solve:     $\Delta x = x_f - x_i = -2\,mi - 3\,mi = -5\,mi$

  • Assess:

  • The displacement is negative ($-5\,mi$), which correctly aligns with westward motion.

  • The magnitude indicates a total position change of $5\,mi$ from start to finish.

  • Physical Sanity Check: Traveling $5\,mi$ in a half-hour ($0.5\,h$) corresponds to a speed of $10\,mph$, which is a highly reasonable pace for a cyclist.