Acceleration

Acceleration Appears When Velocity Changes

When a car starts moving, your body may feel pushed backward. When the car brakes, your body may feel pushed forward. Both moments point to the same physics idea: the car's velocity is changing.

In physics, acceleration describes the change in velocity during a time interval. Since velocity has magnitude and direction, acceleration must also be read together with the positive direction we choose.

Acceleration does not always mean speeding up. Acceleration means velocity changes.

\begin{aligned} \vec{a} &= \frac{\Delta \vec{v}}{\Delta t} \\ &= \frac{\vec{v}_t-\vec{v}_0}{t_t-t_0} \end{aligned}

The unit of acceleration is $\text{m/s}^2$ . If $a=2\text{ m/s}^2$ , the velocity increases by $2\text{ m/s}$ every $1\text{ s}$ as long as the acceleration stays constant.

From Ticker Tape to Graph Slope

A ticker timer marks a paper tape at equal time intervals. If the gaps between marks get wider, the object covers more distance in each equal time interval. Its velocity is increasing. If the gaps stay the same, its velocity is constant. If the gaps get smaller, its velocity is decreasing.

Rocket Acceleration Through Space

A chase camera follows the rocket while

1\text{ s}

time gates spread out, stay even, or bunch together.

Initial velocity: $v_0=2\text{ m/s}$
Acceleration: $a=+4\text{ m/s}^2$
Final velocity: $v_t=18\text{ m/s}$
Ghost interval: $\Delta t=1\text{ s}$

A velocity-time graph reads the same idea in a shorter way. The slope of the line shows how much velocity changes compared with the time it takes.

After the trace is visible, shift attention to the graph. The trace gives the motion a physical feel, while the graph makes the velocity change easier to calculate.

Velocity-Time Graph

Choose one segment on the same graph, then read the line slope as velocity change per second.

On the graph, the dashed line shows the full connected graph. The thick line is the segment being read. If the line rises, velocity increases. If it stays flat, velocity is constant. If it slopes downward, velocity decreases. Acceleration is the ratio of $\Delta v$ to $\Delta t$ .

Reading the Sign of Acceleration

The sign of acceleration cannot be read by itself. It always depends on the positive direction we choose. In this one-dimensional example, let the rightward direction be positive.

Acceleration sign	Velocity change	Reading with right as positive
$a>0$	$v_t>v_0$	Velocity increases in the positive direction
$a=0$	$v_t=v_0$	Velocity stays constant
$a<0$	$v_t<v_0$	Velocity decreases if the object is still moving in the positive direction

The last row is where many students trip up. Negative acceleration does not automatically mean the object is moving left. It means the velocity change points toward the negative side of the chosen axis.

Calculating Velocity Change Each Second

Suppose a car changes velocity from $2\text{ m/s}$ to $18\text{ m/s}$ in $4\text{ s}$ . Its acceleration is:

\begin{aligned} a &= \frac{v_t-v_0}{\Delta t} \\ &= \frac{18-2}{4} \\ &= 4\text{ m/s}^2 \end{aligned}

That result means the velocity increases by $4\text{ m/s}$ each second. The sequence becomes $2\text{ m/s}$ , $6\text{ m/s}$ , $10\text{ m/s}$ , and so on if the acceleration remains constant.

Now compare it with a car whose velocity changes from $18\text{ m/s}$ to $2\text{ m/s}$ in $4\text{ s}$ .

\begin{aligned} a &= \frac{2-18}{4} \\ &= -4\text{ m/s}^2 \end{aligned}

The value $-4\text{ m/s}^2$ means the velocity decreases by $4\text{ m/s}$ each second if the car is still moving in the positive direction.

Zero Acceleration Still Needs Interpretation

If $a=0$ , the object's velocity is not changing. That does not always mean the object is at rest. The object may still move with a constant velocity, such as $10\text{ m/s}$ to the right.

A quick check is to look at velocity, not only acceleration. If $a=0$ and $v=0$ , the object is at rest. If $a=0$ but $v\neq 0$ , the object moves in a straight line with constant velocity.