2.4 Acceleration: Key Concepts & Examples

Acceleration is a change in velocity. This change can involve a modification in the magnitude (speed), the direction, or both aspects of the velocity vector.
Mathematical form: a = \frac{d v}{d t}; for one-dimensional components ax = \frac{\Delta vx}{\Delta t}, \; ay = \frac{\Delta vy}{\Delta t}.
Motion diagram: successive velocity vectors grow farther apart (if speeding up) or closer together (if slowing down); the difference between successive velocity vectors represents the instantaneous acceleration vector.
Constant acceleration: the change in velocity (\Delta v) between equal time intervals is always the same, meaning the velocity vectors change by equal amounts and in the same direction each interval, resulting in a constant acceleration vector.

With a constant-acceleration cart (e.g., fan cart), velocity vectors increase steadily; the acceleration remains constant in both magnitude and direction during each interval. In motion diagrams for constant acceleration, the difference vectors (\vec{v}{i+1} - \vec{v}i) that represent acceleration are all identical.
Velocity vs time graph: the slope of the v-t graph at any point gives the instantaneous acceleration. A higher acceleration (larger magnitude) yields a steeper velocity-time line.
Example: turn the fan to high \u2192 larger acceleration \u2192 steeper velocity-time line.
For a vertical case (e.g., a ball dropped): velocity becomes negative (downward) as it speeds up downward due to gravitational acceleration; the sign and direction of acceleration are determined by how the velocity changes over time.

Sign convention: typically, positive x to the right and positive y upward. Consistency in sign convention is crucial for correct interpretation.
Acceleration direction: This is determined by the direction of the acceleration vector itself, which points in the direction of the change in velocity (\Delta \vec{v}), not inherently by whether the object's speed is increasing or decreasing.
Relationship to speed:
- If velocity and acceleration have the same sign (e.g., both positive or both negative), the object is speeding up.
- If velocity and acceleration have opposite signs (e.g., velocity positive, acceleration negative), the object is slowing down.
- For instance, if a car is moving right (vx > 0) and braking, its acceleration is to the left (ax < 0), causing it to slow down. If it's moving left (vx < 0) and braking, its acceleration is to the right (ax > 0), also causing it to slow down.
Acceleration and velocity-time graph: The sign of the slope of the velocity-time graph directly indicates the sign and direction of the acceleration.
In the elevator example: if the elevator is moving downward (negative velocity) but coming to a stop (speed is decreasing towards zero), the acceleration must be in the opposite direction of motion, meaning it's positive (upward).

Definitions: ax = \frac{\Delta vx}{\Delta t}, \; ay = \frac{\Delta vy}{\Delta t}.
Units: ax, ay \text{ in } \mathrm{m\,s^{-2}}. (Commonly read as meters per second squared.)
Example: time to reach 60 mph.
- 60 mph \u2248 27\ \mathrm{m\,s^{-1}}.
- Corvette: a_x = \frac{27}{3.6} \approx 7.5\ \mathrm{m\,s^{-2}}.
- Sonic: a_x = \frac{27}{9.0} = 3.0\ \mathrm{m\,s^{-2}}.
Concept: acceleration is the slope of the velocity-time graph; the unit is read as meters per second per second (i.e., \mathrm{m\,s^{-2}} ).

Constant-acceleration cars (Table 2.2 context): Corvette reaches 60 mph in 3.6 s \u2192 ax = 7.5\ \mathrm{m\,s^{-2}}; Sonic reaches 60 mph in 9.0 s \u2192 ax = 3.0\ \mathrm{m\,s^{-2}}.
Example 2.6 (lion acceleration): a = 9.5\ \mathrm{m\,s^{-2}}; target speed v_f = 10\ \mathrm{mph} \approx 4.5\ \mathrm{m\,s^{-1}}; time to reach target: \Delta t = \frac{\Delta v}{a} = \frac{4.5}{9.5} \approx 0.47\ \mathrm{s}..
Velocity-time graphs for constant acceleration: straight-line graphs; slope equals acceleration; for the Corvette vs Sonic, the steeper line (Corvette) has the larger acceleration.
Contextual idea: a velocity-time graph can be divided into segments where the slope (acceleration) is constant; middle segment with zero slope means zero acceleration (constant velocity) even if velocity is nonzero.

Acceleration is the rate of change of velocity: a(t) = \frac{dv}{dt}; for one dimension, use ax, ay.
Sign convention: positive to the right/up, negative to the left/down.
Same sign (vx and ax) \u21d2 speeding up; opposite signs \u21d2 slowing down.
The sign of acceleration relates to direction of the acceleration vector, not the instantaneous speed.
Zero acceleration means constant velocity; velocity can be nonzero in this case.
The slope of the