Motion Concepts: Distance, Displacement, Speed, and Velocity (Video Notes)

Distance vs. Displacement

• Distance is a scalar quantity: it only has magnitude, no direction. It represents the total length of the path traveled.
• Displacement is a vector quantity: it has both magnitude and direction. It represents the straight-line change in position from the initial point to the final point.
• Along a chosen axis (e.g., the x-axis):
  • Positive direction is conventionally to the right (or east).
  • Negative direction is to the left (or west).
• Example setup from the transcript:
  • Initial position: $x_i = 100\text{ m}$
  • Final position: $x_f = 20\text{ m}$
  • Displacement along x: $\Delta x = x<em>f - x</em>i = 20 - 100 = -80\ \text{m}$
  • Interpretation: displacement is $80\ \text{m}$ to the left (negative direction).
• Important distinction:
  • Distances are always nonnegative magnitudes (e.g., the total path length).
  • Displacement can be negative or positive depending on the direction relative to the chosen axis.
• Summary from transcript:
  • The displacement has both magnitude and direction, so it is a vector quantity.

Direction and Sign Convention

• Direction is essential for displacement but not for distance.
• In x-axis problems:
  • Right/up/east is typically taken as positive.
  • Left/down/west is typically taken as negative.
• Examples from the transcript reinforce sign conventions:
  • A movement to the right corresponds to a positive value (e.g., +80 m if that were the displacement).
  • A movement to the left corresponds to a negative value (e.g., -80 m).
• Practical note:
  • Always state the chosen coordinate system before solving problems to avoid sign confusion.

Speed

• Definition: speed is how fast an object changes its location.
• It is a scalar quantity (no direction).
• Common units:
  • Meters per second: $v = \frac{D}{T}, \quad \text{with } v \text{ in } \text{m s}^{-1}$
  • Alternative: $v = \frac{D}{T} = \frac{\text{meters}}{\text{seconds}}$
  • In non-SI units, miles per hour (mph) is also common (e.g., 70 mph).
• Example from transcript:
  • If distance is 2 meters traveled in 1 second, then
  • $v = \frac{D}{T} = \frac{2\ \text{m}}{1\ \text{s}} = 2\ \text{m s}^{-1}$

Average Speed

• Definition: average speed is the total distance traveled divided by the total time taken.
• Formula:
  • $\bar{v} = \frac{D}{\Delta t}$
  • Here, $D$ is the total distance traveled (path length) and $\Delta t$ is the total time elapsed.
• Example from transcript:
  • If an object covers 20 meters in 4 seconds:
  • $\bar{v} = \frac{20\ \text{m}}{4\ \text{s}} = 5\ \text{m s}^{-1}$

Velocity

• Definition: velocity is the displacement per unit time; it contains both magnitude and direction.
• It is a vector quantity:
  • $\vec{v} = \frac{\Delta \mathbf{r}}{\Delta t}$
  • Where $\Delta \mathbf{r}$ is the displacement vector from initial to final position, and $\Delta t$ is the time interval.
• Directional example:
  • If a runner moves 200 meters east, the displacement is +200 m in the +x direction, and the velocity depends on the time interval used.
• Relationship to speed:
  • Speed is the magnitude of velocity when direction is ignored:
  • $|\,\vec{v}\,| = v = \frac{D}{\Delta t} \quad\text{(if using speed as the magnitude)}$

Examples and Scenarios (Key Applications)

• Scenario 1 (displacement on x-axis):
  • Start at $x<em>i = 100\text{ m}$, end at $x</em>f = 20\text{ m}$
  • Displacement: $\Delta x = x<em>f - x</em>i = -80\text{ m}$
  • Magnitude of displacement: $|\Delta x| = 80\text{ m}$; direction: left (negative).
• Scenario 2 (distance and speed):
  • Distance traveled: $D = 2\text{ m}$ in $T = 1\text{ s}$
  • Speed: $v = \frac{D}{T} = \frac{2\text{ m}}{1\text{ s}} = 2\text{ m s}^{-1}$
• Scenario 3 (average speed over a path with a non-straight path):
  • Path length $D$ is used for speed, not the straight-line displacement.
  • If the total distance is 20 m and total time is 4 s, then the average speed is $5\ \text{m s}^{-1}$ as shown above.
• Scenario 4 (velocity against a displacement example):
  • A runner covers $200\text{ m}$ east as the displacement.
  • The velocity vector over the time interval is $\vec{v} = (200\text{ m})/\Delta t$ in the +x (east) direction.
• Conceptual point on changing velocity:
  • Velocity can be changed by changing speed (magnitude), changing direction, or both.
  • To alter velocity while keeping the same direction, you would change the speed; to alter direction, you would rotate the velocity vector.

Connections to Foundational Principles

• Distinction between scalar and vector quantities:
  • Distance is scalar; displacement is vector.
  • Speed is scalar; velocity is vector.
• Sign conventions and coordinate systems are essential for resolving directions in one-dimensional motion along an axis.
• The relationship between path length (distance) and straight-line change in position (displacement) underpins the difference between speed and velocity.

Formulas and Key Equations (LaTeX)

• Displacement (vector):
  • $\Delta \mathbf{r} = \mathbf{r}(t<em>f) - \mathbf{r}(t</em>i)$
  • For one-dimensional motion along x-axis: $\Delta x = x<em>f - x</em>i$
• Distance (magnitude of path length): no sign, always nonnegative
• Speed (magnitude of velocity):
  • $v = \frac{D}{T}$
• Average speed:
  • $\bar{v} = \frac{D}{\Delta t}$
• Velocity (vector):
  • $\vec{v} = \frac{\Delta \mathbf{r}}{\Delta t}$
• Units:
  • Speed/velocity units: $\text{m s}^{-1}$ (meters per second)
  • Alternative notation: $v = \frac{\text{m}}{\text{s}}$

Practical Notes for Problem-Solving

• Always identify whether you are dealing with distance (path length) or displacement (net change in position).
• Decide on a consistent coordinate system before computing signs and directions.
• Distinguish between speed (magnitude only) and velocity (magnitude and direction).
• Use the formula $\bar{v} = D / \Delta t$ for average speed when given total distance and total time.
• Use $\vec{v} = \Delta \mathbf{r} / \Delta t$ to determine velocity; pay attention to both magnitude and direction of displacement.