3.1 Acceleration

Acceleration in One-Dimensional Motion

Acceleration describes how quickly an object's velocity changes over time. It connects force, motion, and the kinematic equations you'll use throughout physics, so building a solid understanding here pays off in nearly every unit that follows.

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Direction and Magnitude of Acceleration

Acceleration measures the rate of change of velocity over time, expressed in units of $m/s^2$. Because acceleration is a vector quantity, it has both magnitude and direction.

Acceleration can be positive, negative, or zero:

• Positive acceleration: velocity is increasing (in the positive direction) or decreasing (in the negative direction)
• Negative acceleration: velocity is decreasing (in the positive direction) or increasing (in the negative direction)
• Zero acceleration: velocity stays constant

A common mistake is thinking "negative acceleration" always means "slowing down." That's only true if the object is moving in the positive direction. If an object moves to the left (negative direction) and has negative acceleration, it's actually speeding up. The key rule: when velocity and acceleration point the same direction, the object speeds up. When they point in opposite directions, the object slows down.

Analysis with Kinematic Equations

The kinematic equations describe motion with constant acceleration using five variables: displacement ($\Delta x$), initial velocity ($v_0$), final velocity ($v_f$), acceleration ($a$), and time ($t$).

• $v_f = v_0 + at$ — use when you don't need displacement
• $\Delta x = v_0 t + \frac{1}{2}at^2$ — use when you don't know final velocity
• $v_f^2 = v_0^2 + 2a\Delta x$ — use when time isn't given

Each equation is missing one of the five variables. To solve a problem, identify which three variables you know, which one you want, and pick the equation that leaves out the fifth.

Graphs and motion analysis:

• On a position-time graph, the slope at any point equals the velocity. A straight line means constant velocity (zero acceleration). A curve means the velocity is changing, so acceleration is present.
• On a velocity-time graph, the slope equals the acceleration, and the area under the curve equals the displacement. A horizontal line means zero acceleration; a sloped line means constant acceleration.

Types of Acceleration Calculations

Average acceleration gives the overall rate of velocity change across a time interval:

$a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_0}{t_f - t_0}$

For example, if a car goes from $5 \, m/s$ to $25 \, m/s$ in $4 \, s$, the average acceleration is $\frac{25 - 5}{4} = 5 \, m/s^2$.

Instantaneous acceleration is the acceleration at one specific moment. On a velocity-time graph, you find it by drawing a tangent line at that point and calculating its slope. If the velocity-time graph is a straight line (constant acceleration), the instantaneous acceleration equals the average acceleration everywhere on that interval.

Forces and Acceleration

Newton's second law ties force directly to acceleration:

$a = \frac{F_{net}}{m}$

This tells you two things. First, acceleration is directly proportional to net force: double the force, double the acceleration. Second, acceleration is inversely proportional to mass: double the mass with the same force, and acceleration is cut in half. Inertia, an object's resistance to changes in motion, is measured by mass. More massive objects are harder to accelerate.

Two special cases worth noting:

• Free fall: Near Earth's surface, all objects experience a constant downward acceleration of approximately $9.8 \, m/s^2$ (often rounded to $10 \, m/s^2$ in problems), regardless of mass, when air resistance is negligible.
• Centripetal acceleration: In circular motion, acceleration points toward the center of the circle, continuously changing the direction of velocity even if speed stays constant. You'll explore this more in later units.