One-Dimensional Motion: Comprehensive Notes

One-Dimensional Motion at Constant Velocity

• Introduction to kinematics: study of motion, focusing here on motion descriptions rather than causes (dynamics discussed later).

  • This pre-lecture covers straight-line motion at constant velocity.
  • Real-world examples: a blood cell moving along a capillary; a swimmer racing down a pool.
  • In a subsequent pre-lecture, we expand to accelerated straight-line motion (e.g., an airplane accelerating down a runway).

• Learning objectives for this pre-lecture:

  • Explain what is meant by linear motion (motion along a straight line).
  • Understand the meanings and relationships among displacement, average velocity, and constant velocity.
  • Solve problems involving linear motion at constant velocity.

• Coordinate system and position:

  • We describe the motion of an object using a coordinate system with position denoted by $x$ as a function of time $t$.
  • The origin, or reference position, is defined as $x = x_0 = 0$ in the chosen coordinate system.
  • The coordinate system defines the positive direction; an object's position is positive if it lies on the positive side of the origin, and negative if on the negative side.
  • In the SI system, position is measured in meters (m).
  • Position is a vector: it has both a magnitude (size) and a direction.

• Example: a 50 meter pool with origin at the midpoint and positive direction to the right:

  • If you stand at $x = +10 \text{ m}$, you are 10 m to the right of the origin.
  • If your friend stands at $x = -10 \text{ m}$, he is 10 m to the left of the origin.
  • The sign of the position identifies the location relative to the origin.
  • Distance vs displacement:
  • Distance is a scalar: it has magnitude but no direction.
  • Displacement is a vector: it includes both magnitude and direction, calculated from the initial and final positions.

• Displacement and example with swimmers:

  • Consider the midpoint of the pool (the origin is at the pool center).
  • To reach the midpoint from either end, each swimmer must swim 25 m.
  • If one swimmer moves to the right (positive x) and the other to the left (negative x), their displacements differ even though the distances traveled are the same.
  • Mathematically, displacement is: $\Delta x = x - x<em>0$ where $x</em>0$ is the initial position and $x$ is the final position.
  • For the swimmer moving to the right toward the center, displacement is positive (+25 m); for the swimmer moving to the left, displacement is negative (−25 m).

• Time and scalar nature:

  • Time is a scalar quantity (no direction): it has magnitude only.
  • It is the interval over which motion occurs.

• Speed vs velocity:

  • Average speed = total distance traveled / time interval.
  • If a swimmer travels 50 m in 25 s, his average speed is
    $\text{speed}_{\text{avg}} = \frac{\text{distance}}{\text{time}} = \frac{50\ \text{m}}{25\ \text{s}} = 2\ \text{m s}^{-1}.$
  • Speed is a scalar; it has magnitude only and no direction.
  • Average velocity = displacement / time interval.
  • Because displacement has direction, average velocity is a vector.
  • It includes a magnitude and a direction along the x-axis (denoted by a subscript, e.g., $v_x$).
  • For the same swimmer, if the displacement is +25 m and the time is 50 s, $v_{\text{avg}} = \Delta x / \Delta t = +\frac{25}{50} = +0.5\ \text{m s}^{-1}.$
  • If the swimmer travels the 25 m to the left in 12.5 s, the velocity is negative: $v_{\text{avg}} = -2.0\ \text{m s}^{-1}$.

• Delta notation and sign convention:

  • The triangle symbol (capital Delta) denotes a change: $\Delta$.
  • $\Delta x$ is the change in position along the x-axis and is calculated as $\Delta x = x - x_0$.
  • $\Delta t$ is the change in time: $\Delta t = t - t_0$.
  • When we talk about velocity along the x-axis, we write $v<em>x$; for other directions, we use $v</em>y, v_z$ as needed.

• Visual representations of one-dimensional motion:

  • Motion diagrams: dots plotted at equal time intervals represent the position of an object at those times. Faster motion yields more widely spaced dots for the same elapsed time.
  • Position-time graphs: position on the vertical axis, time on the horizontal axis. A 90-degree rotation of the motion diagram can help visualize this graph as a function of time.
  • Construction of a position-time graph (sketch): rotate the diagram so that the positive x-axis points upward, place each dot at the corresponding time along the horizontal axis, and connect the points with a straight line.

• Relationship between position, velocity, and time for constant velocity motion:

  • For constant velocity, the average velocity equals the instantaneous velocity at any time, i.e., the velocity is constant.
  • Let the motion start at time $t<em>0 = 0$ with initial position $x</em>0$ and final position $x$ after time $t$.
  • The position of the object at any time t is given by:$x = x<em>0 + v</em>x t.$
  • This equation allows calculation of the position at any time for constant velocity, given the starting position and velocity.

• Example problem: two swimmers with constant velocities meet along the pool

  • Setup: Swimmer A (to the right) moves at $v<em>A = +0.5\ \text{m s}^{-1}$ and Swimmer B (to the left) moves at $v</em>B = -2.0\ \text{m s}^{-1}$.
  • Initial positions: Swimmer A starts at $x<em>A(0) = -25\ \text{m}$ (left end), Swimmer B starts at $x</em>B(0) = +25\ \text{m}$ (right end).
  • Position as a function of time:
  • $x<em>A(t) = x</em>{A0} + v_A t = -25 + 0.5 t.$
  • $x<em>B(t) = x</em>{B0} + v_B t = +25 - 2 t.$
  • They meet when $x<em>A(t) = x</em>B(t)$:
  • Solve: $-25 + 0.5 t = 25 - 2 t$ → $2.5 t = 50$ → $t = 20\ \text{s}.$
  • Meeting position: plug back into either equation, e.g., $x_A(20) = -25 + 0.5(20) = -25 + 10 = -15\ \text{m}.$
  • Check with swimmer B: $x_B(20) = 25 - 2(20) = 25 - 40 = -15\ \text{m}.$
  • Graphical interpretation: the lines of $x<em>A(t)$ and $x</em>B(t)$ cross at $t = 20\ \text{s}$ and $x = -15\ \text{m}.$

• Key takeaways and relationships:

  • Linear motion refers to motion along a straight line; constant velocity means velocity does not change with time.
  • Position is a vector; displacement is the change in position (a vector).
  • Distance traveled is a scalar; it only has magnitude, no direction.
  • Average speed uses distance / time; average velocity uses displacement / time and is a vector.
  • The subscript on velocity (e.g., $v<em>x$) indicates the component along a particular axis; a higher-dimensional motion would use $v</em>y, v_z$ as needed.
  • A useful shorthand for changes is $\Delta$: $\Delta x = x - x<em>0\; ,\; \Delta t = t - t</em>0.$
  • If a position-time graph shows a line with a negative slope, the average velocity is negative along the chosen axis, indicating motion in the negative direction.
  • The slope of a position-time graph between two times gives the average velocity over that interval:
  • $v<em>{\text{avg},x} = \frac{\Delta x}{\Delta t}$ where $\Delta x = x</em>2 - x<em>1$ and $\Delta t = t</em>2 - t_1$.
  • In one dimension with constant velocity, the following relationship holds for all times:$x(t) = x<em>0 + v</em>x t$, given that you start counting time at $t = 0$ and know the initial position $x<em>0$ and velocity $v</em>x$.

• Notation, conventions, and practical implications:

  • The origin can be chosen anywhere convenient for a problem, and the positive direction is defined accordingly.
  • Position is a vector along the chosen axis; the velocity component along the axis is denoted as $v<em>x$ (and similarly for $v</em>y, v_z$).
  • The concept of delta notation is essential for expressing changes and relating position, velocity, and time.
  • These concepts provide the foundation for solving linear motion problems and for interpreting motion diagrams and graphs in one dimension.