Acceleration Study Notes

An object in uniform motion moves along a straight line with an unchanging velocity.
Nonuniform motion is more common, where velocity is changing.
Examples of nonuniform motion include:
- Balls rolling down hills.
- Cars braking to a stop.
- Falling objects.
Later modules will address nonuniform motion not confined to a straight line, such as:
- Circular motion.
- The motion of thrown objects, like baseballs.

Uniform motion feels smooth; one does not feel motion when moving uniformly.
Non-uniform motion (e.g., turning or going over a roller coaster hill) induces feelings of being pushed or pulled.
Figure 1 includes motion diagrams:
- Diagram 1: Indicates motionless (e.g., runner waiting).
- Diagram 2: Distances between positions are constant (uniform motion at constant velocity).
- Diagram 3: Increasing distance indicates the jogger is speeding up.
- Diagram 4: Decreasing distance indicates the jogger is slowing down.

Definition: A particle model motion diagram displays an object with changing velocity.
- Indicates changes in velocity via the spacing of dots and lengths of velocity vectors.
Characteristics:
- If an object speeds up:
- Each successive velocity vector is longer.
- Spacing between dots increases.
- If the object slows down:
- Each velocity vector is shorter.
- Spacing between dots decreases.

Acceleration is defined as the rate at which an object's velocity changes.
Including acceleration vectors in motion diagrams provides a full picture of movement.
Acceleration vector is calculated:
- Find change in velocity: $riangle v = vf - vi$
- Average over time interval: $a = \frac{ riangle v}{ riangle t}$ where $riangle t$ is the time interval.

Use the velocity vectors to draw the acceleration vector:
- Start with the final velocity vector $vf$ , then draw initial velocity vector $vi$ .
- Draw acceleration vector $a$ from tail of $vi$ to tip of $vf$ .
For constant acceleration, length and direction of the vector are determined by:
- $a = \frac{ riangle v}{ riangle t}$

Figure 4 illustrates four scenarios:
1. Car speeding up in positive direction – Velocity and acceleration vectors point in the same direction.
2. Car slowing down in positive direction – Velocity and acceleration vectors are opposite.
3. Car speeding up in negative direction – Both vectors align in the negative direction.
4. Car slowing down in negative direction – Vectors are in opposing directions.
Important points:
- Positive acceleration means acceleration vector points positively.
- Negative acceleration indicates velocity vector points negatively.
Observation: Sign of acceleration alone does not indicate speeding up or slowing down. Both velocity and acceleration directions are needed for analysis.

Definition: A velocity-time graph plots velocity against time (velocity on vertical axis, time on horizontal axis).
Example: Car accelerating from rest.
- Graph shows a straight line indicating constant acceleration.
- Acceleration calculated by slope: $ext{slope} = \frac{ ext{rise}}{ ext{run}}$ .
- For example, a slope of $5.00 ext{ m/s}^{2}$ indicates a change in velocity of 5.0 m/s in 1.0 seconds.
Five runners’ motions shown in Figure 6:
- Different graphs indicate varying accelerations and velocities.

Average Acceleration: Change in velocity during a specific time interval divided by that interval.
Units: Measured in $ext{m/s}^2$ .
Instantaneous acceleration calculated using tangent line on a velocity-time graph.
For constant acceleration, average and instantaneous accelerations are equal.

Calculating average acceleration:
If a car accelerates from 4.0 m/s to 36 m/s in 4.0 s:
- Average acceleration = $\frac{36 ext{ m/s} - 4 ext{ m/s}}{4.0 ext{ s}} = 8 ext{ m/s}^2$ .
Average acceleration of a bus stopping from 25 m/s to 0 m/s in 3.0 s is:
- $a = \frac{0 ext{ m/s} - 25 ext{ m/s}}{3 ext{ s}} = -8.33 ext{ m/s}^2$ .

Acceleration is a matter of both speed change and direction change. Understanding its implications is crucial for analyzing motion dynamics.