2.4 Acceleration

Acceleration and Velocity

Acceleration measures how quickly velocity changes over time. Since velocity includes both speed and direction, acceleration can involve changes to either one (or both). Understanding acceleration is the key link between describing motion and explaining why things move the way they do.

Acceleration and Velocity Changes

Acceleration is the rate of change of velocity. That means any time an object's velocity changes, acceleration is happening. A few ways this can occur:

• The object speeds up (magnitude of velocity increases)
• The object slows down (magnitude of velocity decreases)
• The object changes direction, even if its speed stays the same

A car merging onto a highway speeds up. A skydiver deploying a parachute slows down. A satellite in a circular orbit around Earth maintains constant speed but continuously changes direction, so it's still accelerating.

Positive vs. negative acceleration depends on your chosen coordinate system, not on whether something is "speeding up" or "slowing down":

• Positive acceleration means the velocity is becoming more positive over time. A rocket launching upward (if up is positive) has positive acceleration. So does a car that's reversing (negative velocity) and hitting the brakes, because its velocity is shifting toward zero (becoming less negative, i.e., more positive).
• Negative acceleration means the velocity is becoming more negative over time. A car driving forward and braking has negative acceleration. So does a car in reverse that's pressing the gas pedal harder, because its velocity is becoming more negative.

The core rule: when velocity and acceleration share the same sign, the object speeds up. When they have opposite signs, the object slows down.

Calculation of Acceleration

Average acceleration is calculated with this formula:

$a = \frac{v_f - v_i}{t}$

• $v_f$ = final velocity
• $v_i$ = initial velocity
• $t$ = time interval

The unit for acceleration is meters per second squared ($m/s^2$), which you can read as "meters per second, per second." That tells you how many $m/s$ the velocity changes each second.

Example: A car accelerates from 20 m/s to 30 m/s in 5 seconds.

1. Identify your values: $v_i = 20 \, m/s$, $v_f = 30 \, m/s$, $t = 5 \, s$

2. Plug into the formula: $a = \frac{30 \, m/s - 20 \, m/s}{5 \, s}$

3. Solve: $a = \frac{10 \, m/s}{5 \, s} = 2 \, m/s^2$

The velocity increased by 2 m/s every second.

Instantaneous acceleration is the acceleration at a single moment in time. Conceptually, it's what you get when the time interval shrinks toward zero. On a velocity vs. time graph, instantaneous acceleration is the slope of the tangent line at that point.

Signs of Velocity and Acceleration

This is where students often get tripped up. The sign of velocity and the sign of acceleration tell you different things:

• Velocity sign → direction of motion (positive = right/up, negative = left/down, based on your coordinate choice)
• Acceleration sign → direction the velocity is changing

Combining them:

Notice that negative acceleration does not automatically mean slowing down. A negative acceleration on a negative velocity actually speeds the object up. Always compare the signs.

Forces and Motion

Acceleration doesn't just happen on its own. A net force is required to cause it. This connection is formalized in Newton's Second Law, which you'll study in detail later:

$F_{net} = ma$

This tells you that the acceleration of an object is directly proportional to the net force on it and inversely proportional to its mass. A larger force produces more acceleration; a larger mass resists acceleration more.

For now, the important distinction is that kinematics (this unit) describes how objects move using displacement, velocity, and acceleration. It doesn't ask why. The "why" comes from forces, which is the subject of dynamics. Acceleration is the concept that bridges the two.