Comprehensive Notes on One-Dimensional Motion and Kinematics

Foundations of Kinematics and Reference Frames

Motion in one dimension begins with kinematics, which is defined as the study of motion without considering its causes. Understanding motion requires the identification of a reference frame, which is the coordinate system from which the positions of objects are described. Motion is relative to the observer; for example, passengers on a train like the Shanghai Maglev traveling near $300\,\text{mph}$ might not notice they are moving unless they look out the window. At such speeds, the $439\,\text{mile}$ trip from Boston to Washington, DC, could be completed in under an hour and a half, whereas the fastest current trains take over six hours.

Position is the location of an object at any particular time, always specified relative to a convenient reference frame. While Earth is often used as a stationary reference frame, others may be in motion relative to Earth. To describe a person's position on an airplane, the airplane itself serves as the reference frame rather than the ground. A description of motion is only meaningful when a reference frame is specified. This concept was explored in the 17th century by Galileo Galilei through a thought experiment involving a windowless ship moving at a constant speed and direction on a calm sea. Galileo concluded that a person inside the ship and a person on the shore would describe each other's motion symmetrically and oppositely, and that observers moving at a constant speed relative to each other describe motion in the same way.

Position, Distance, and Displacement

Describing motion requires precise variables for position. The variable $d$ represents an object's position, with subscripts used to distinguish the starting point, or initial position ($d_0$), from the final position ($d_f$). The term $d_0$ is pronounced "d naught," where the zero stands for initial. In some contexts, $x$ or $s$ may be substituted for $d$. In two-dimensional motion, subscripts like $d_x$ (horizontal) or $d_y$ (vertical) are used.

Distance is defined as the length of the path actually traveled between an initial and a final position. It is a scalar quantity, meaning it has magnitude—the size or amount—but no direction. Conversely, displacement is the net change in position of an object against a fixed axis, defined as a vector quantity. Vectors possess both magnitude and direction. Displacement is calculated by subtracting the original position from the final position using the formula $\Delta d = d_f - d_0$. The Greek letter delta ($\Delta$) signifies "change in."

In a round trip where an object returns to its starting point, the total displacement is $0$, even if the distance traveled is significant. For instance, if a parent drives $5\,\text{km}$ to school and then $5\,\text{km}$ back home, the total distance is $10\,\text{km}$, but the displacement is $0\,\text{km}$. If the forward direction is assigned a positive value ($+$) and the opposite direction a negative value ($-$), the two portions of the trip cancel each other out.

Mathematical Application and SI Units

Mathematically, displacement requires an axis with a defined origin ($O$) and designated positive and negative directions. Choosing a logical origin, such as starting at zero, often simplifies calculations and avoids negative signs. In physics, standard units known as SI units (International System of Units), based on the metric system, are preferred. The standard unit for displacement and distance is the meter ($\text{m}$). Consistency in units is vital; failure to convert between English units and SI units resulted in the 1998 loss of NASA's Mars Climate Orbiter, a $125\,\text{million}$ dollar satellite that disintegrated in the Martian atmosphere after orbiting too low at $187,000\,\text{feet}$.

Calculating displacement involves specific steps. If a cyclist rides $3\,\text{km}$ west and then $2\,\text{km}$ east, and east is defined as the positive direction, the final displacement is $\Delta d = (-3\,\text{km}) + 2\,\text{km} = -1\,\text{km}$, or $1\,\text{km}$ west. The distance traveled, however, is simply the sum of the magnitudes: $3\,\text{km} + 2\,\text{km} = 5\,\text{km}$. The magnitude of the displacement is the absolute value of the displacement vector, which in this case is $1\,\text{km}$.

Speed and Velocity

Speed is the rate at which an object changes its location and is a scalar quantity. Average speed ($v_{\text{avg}}$) is calculated as the total distance traveled divided by the time elapsed ($\Delta t$), where $\Delta t = t_f - t_0$. The SI unit for speed is meters per second ($\text{m/s}$). Instantaneous speed refers to the speed at a specific instant in time, such as the reading on a car's speedometer. The relationship can be rearranged as $d = v_{\text{avg}} \times \Delta t$ or $\Delta t = \frac{d}{v_{\text{avg}}}$.

Velocity is the vector version of speed, describing both speed and direction. Average velocity ($v_{\text{avg}}$) is the displacement divided by the time over which it occurs: $v_{\text{avg}} = \frac{\Delta d}{\Delta t}$. In text, vector variables are often bolded ($v$) or topped with an arrow to distinguish them from scalar variables. Like displacement, average velocity can be zero if the net displacement is zero, regardless of the distance covered. For example, a car driving a $6\,\text{km}$ round trip in $30\,\text{minutes}$ has an average speed of $12\,\text{km/h}$, but an average velocity of $0$. Instantaneous velocity is the velocity at a specific moment. If velocity is constant, instantaneous velocity equals average velocity.

Graphical Analysis of Motion: Position vs. Time

Graphs provide numerical information and reveal relationships between physical quantities. In a position vs. time graph, time is typically the independent variable on the horizontal axis ($x$-axis), while position is the dependent variable on the vertical axis ($y$-axis). A straight-line graph follows the linear form $y = mx + b$. In the context of motion, the slope ($m$) represents the average velocity, as it signifies $\frac{\text{rise}}{\text{run}}$ or $\frac{\Delta d}{\Delta t}$. The $y$-intercept ($b$) represents the initial position ($d_0$).

If the graph is a curve, the velocity is changing over time. The steeper the slope, the greater the velocity. To find the instantaneous velocity at any specific point on a curved position-time graph, one must draw a tangent line to the curve at that point and calculate the slope of the tangent line. A tangent is a line that touches the curve at exactly one point. Average velocity for a complicated trip consisting of different segments can be calculated as a weighted average of the velocities of those segments.

Graphical Analysis of Motion: Velocity vs. Time

Velocity vs. time graphs allow for the determination of both displacement and acceleration. Acceleration is defined as the rate of change of velocity: $a = \frac{\Delta v}{\Delta t}$. On a velocity vs. time graph, the slope of the line represents the acceleration. A horizontal line indicates constant velocity and zero acceleration. The general equation for a linear velocity-time graph is $v = v_0 + at$, where $v_0$ is the initial velocity ($y$-intercept) and $a$ is the acceleration (slope).

The area under the curve in a velocity vs. time graph represents the displacement ($\Delta d$). For a shape consisting of a rectangle, the area is simply $\text{length} \times \text{width}$. For a triangle, the area is $0.5 \times \text{base} \times \text{height}$. Adding these areas provides the net displacement. For instance, in a jet car example where the car reaches $160\,\text{m/s}$ over $30\,\text{seconds}$, the displacement is calculated by summing the area of a rectangle ($20\,\text{m/s} \times 30\,\text{s} = 600\,\text{m}$) and a triangle ($0.5 \times 30\,\text{s} \times 140\,\text{m/s} = 2,100\,\text{m}$) to reach a total of $2,700\,\text{m}$.

Dimensional analysis is a tool used to verify the correctness of physical calculations by treating units like numbers. For example, multiplying velocity ($\text{m/s}$) by time ($\text{s}$) yields meters ($\text{m}$), confirming the formula for displacement is correct. If the velocity graph is a more realistic, undefined curve, displacement must be estimated by breaking the curve into smaller intervals and estimating the area of each section using geometric approximations.

Questions & Discussion

Question: Are clouds a useful reference frame for airplane passengers? Why or why not?

Answer: Generally, clouds are not a useful reference frame because they are often in motion themselves relative to the Earth. A useful reference frame is ideally stationary or has a known, constant speed and direction.

Question: Imagine standing on a platform watching a train pass by. How would your description of motion compare to that of a person on the train?

Answer: You would see the train moving past you, while the person on the train would see you moving past the train in the opposite direction.

Question: How do different reference frames affect the description of a bouncing ball when the person bouncing it is walking forward?

Answer: The motion of the ball is dependent on the reference frame and is different for different reference frames. To a stationary observer, the ball moves in a zigzag or parabolic path. To the person walking with the ball, it appears to move straight up and down.

Question: If a person takes three steps and ends up in the exact same place as their starting point, what must be true?

Answer: The total displacement is zero, and therefore the average velocity for the entire movement must also be zero.

Question: Can average velocity be negative?

Answer: Yes, average velocity is negative if the net displacement is negative, indicating motion in the direction defined as negative on the coordinate axis.