KAHIBARO
Table of Contents
Meaning of Acceleration
Acceleration describes how velocity changes with time. In one dimensional motion, velocity can change in two ways. Its magnitude can change, meaning the object speeds up or slows down, and its sign can change, meaning the direction of motion can change along the chosen axis.
If position is measured along an $x$ axis, then acceleration tells us how quickly the velocity $v$ changes as time passes. Because velocity is a vector quantity, acceleration also has a direction. In one dimension, that direction is represented by a positive or negative sign.
Average Acceleration
Over a time interval from $t_1$ to $t_2$, the average acceleration is the change in velocity divided by the change in time:
$$
a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1}
$$
This formula tells us the overall rate of change of velocity during the interval. It does not say what happened at every instant in between.
For example, if a car's velocity changes from $4 \,\text{m/s}$ to $10 \,\text{m/s}$ in $3 \,\text{s}$, then
$$
a_{\text{avg}} = \frac{10 - 4}{3} = 2 \,\text{m/s}^2
$$
This means the velocity increased by $2 \,\text{m/s}$ every second on average.
Important formula:
$$
a_{\text{avg}} = \frac{\Delta v}{\Delta t}
$$
Acceleration is the rate of change of velocity, not the rate of change of position.
Instantaneous Acceleration
Sometimes we want the acceleration at a particular moment, not just over a whole interval. This is called instantaneous acceleration. It is defined as the derivative of velocity with respect to time:
$$
a = \frac{dv}{dt}
$$
Since velocity itself is the derivative of position,
$$
v = \frac{dx}{dt}
$$
acceleration can also be written as
$$
a = \frac{d^2 x}{dt^2}
$$
This means acceleration is the second derivative of position with respect to time.
Units of Acceleration
The SI unit of acceleration is meters per second squared, written as $\text{m/s}^2$.
This unit means that the velocity changes by a certain number of meters per second during each second.
For instance, an acceleration of $3 \,\text{m/s}^2$ means the velocity increases by $3 \,\text{m/s}$ every second.
Sign of Acceleration
In one dimension, the sign of acceleration depends on the direction of the chosen coordinate axis. A positive acceleration points in the positive direction, and a negative acceleration points in the negative direction.
It is important not to confuse negative acceleration with slowing down. Negative acceleration does not always mean the object is losing speed. It only means the acceleration points in the negative direction.
Whether an object speeds up or slows down depends on the signs of velocity and acceleration together.
| Velocity | Acceleration | What happens to speed |
|---|---|---|
| Positive | Positive | Speed increases |
| Positive | Negative | Speed decreases |
| Negative | Negative | Speed increases |
| Negative | Positive | Speed decreases |
A negative acceleration does not automatically mean slowing down.
An object slows down when velocity and acceleration have opposite signs.
Examples of Interpretation
Suppose a particle moves to the right, so its velocity is positive. If its acceleration is also positive, it moves faster to the right. If its acceleration is negative, it still moves to the right at that moment, but its speed decreases.
Now suppose the particle moves to the left, so its velocity is negative. If its acceleration is negative, it speeds up to the left. If its acceleration is positive, it slows down.
This is why the sign of velocity matters just as much as the sign of acceleration.
Acceleration as a Slope on a Graph
On a velocity versus time graph, acceleration is the slope.
If the graph is a straight line rising upward, the acceleration is positive. If the graph slopes downward, the acceleration is negative. If the graph is horizontal, the acceleration is zero.
For average acceleration between two times, we use the slope of the secant line. For instantaneous acceleration at one time, we use the slope of the tangent line.
Zero and Changing Acceleration
If acceleration is zero, velocity stays constant. The object may still be moving, but its velocity does not change.
If acceleration changes with time, then the motion becomes more complicated. The definition of acceleration still stays the same, but the value of $a$ is no longer constant.
For example, a runner starting from rest may have a large acceleration at first and then a smaller acceleration later. The acceleration describes how the velocity is changing at each moment.
Relation to Force
In later mechanics, acceleration becomes especially important because forces produce acceleration. For now, in kinematics, we simply describe motion without asking what causes it. Acceleration is one of the main quantities used to describe that motion.
Summary Equations
Key definitions:
$$
a_{\text{avg}} = \frac{v_2 - v_1}{t_2 - t_1}
$$
$$
a = \frac{dv}{dt}
$$
$$
a = \frac{d^2x}{dt^2}
$$
SI unit:
$$
\text{m/s}^2
$$
Acceleration is the measure of how velocity changes with time. In one dimensional motion, understanding its sign is essential. Positive or negative acceleration tells us direction, while speeding up or slowing down depends on the combination of acceleration and velocity.