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

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 = xf - xi = 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 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.

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}

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}

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)}

Scenario 1 (displacement on x-axis):
- Start at $xi = 100\text{ m}$, end at $xf = 20\text{ m}$
- Displacement: \Delta x = xf - xi = -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.

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.

Displacement (vector):
- \Delta \mathbf{r} = \mathbf{r}(tf) - \mathbf{r}(ti)
- For one-dimensional motion along x-axis: \Delta x = xf - xi
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}}

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.