This unit is about acceleration, the rate at which an object's velocity changes, and how to describe it with numbers, equations, and graphs. By the end, you should be able to calculate acceleration, tell whether an object is speeding up or slowing down, solve constant-acceleration problems with the kinematic equations, and read motion graphs using slope and area.
Read the overview first, then work through each topic's notes and try the checkpoint question before checking the answer.
In earlier units you described motion with constant velocity. Now velocity can change, and acceleration measures how quickly it does. This unit gives you the tools to predict where an object will be and how fast it will move when acceleration stays constant.
Honors Physics Unit 3 covers acceleration in one dimension. You learn that average acceleration is and that acceleration is measured in m/s². You also learn how the signs of velocity and acceleration show whether an object speeds up or slows down, and how free fall works with m/s² downward. The second half covers the three constant-acceleration kinematic equations and how to choose between them. It also covers position-time, velocity-time, and acceleration-time graphs, where slopes and areas connect one graph to the next.
An acceleration of 4 m/s² means velocity changes by 4 m/s every second. Because velocity has a direction, acceleration happens whenever speed changes, direction changes, or both. In one-dimensional problems, the sign (+ or −) carries the direction.
When velocity and acceleration have the same sign, the object speeds up. When they have opposite signs, it slows down. A negative acceleration alone does not mean slowing down.
The kinematic equations link displacement, initial velocity, final velocity, acceleration, and time, and each one leaves out a different variable. On graphs, the slope of position-time is velocity, and the slope of velocity-time is acceleration. Area under velocity-time is displacement, and area under acceleration-time is change in velocity.
Every kinematics problem in this unit can be solved with an equation or read from a graph, and both give the same answer. For example, the area under a velocity-time graph equals the displacement you would get from . If you get comfortable moving between the two, you can check your own work.
Acceleration is the rate at which velocity changes, measured in m/s².
Three kinematic equations describe motion with constant acceleration:
Open this guide for a closer review of the topic.
Acceleration is the rate at which velocity changes, measured in m/s². Average acceleration over an interval is . If acceleration is constant at 3 m/s², velocity changes by 3 m/s every second. Acceleration is a vector with a direction.
The sign of acceleration tells you its direction. It does not tell you whether the object is speeding up. To find that out, compare the acceleration's sign with the velocity's sign. If the signs match, the object speeds up, and if they are opposite, it slows down. A car moving left (negative velocity) with negative acceleration is speeding up.
Average acceleration covers a whole time interval. Instantaneous acceleration is the value at one moment, which you find from the slope of a tangent line on a velocity-time graph. When acceleration is constant, the two values are equal.
Near Earth's surface, an object in free fall with negligible air resistance has m/s² downward, whatever its mass. Newton's second law, , previews why forces cause acceleration. You'll use it fully in the forces unit.
Worked example. You throw a ball straight up at 15 m/s. Take up as positive, so m/s². After 2.0 s, m/s. The negative sign means the ball has already passed its highest point and is moving down at 4.6 m/s.
A bike moving at −6 m/s has a constant acceleration of +2 m/s². Is it speeding up or slowing down, and what is its velocity 2 s later? (Answer: It is slowing down because the signs are opposite. Its velocity is m/s.)
| Velocity sign | Acceleration sign | What happens |
|---|---|---|
| Positive | Positive | Speeds up |
| Positive | Negative | Slows down |
| Negative | Negative | Speeds up |
| Negative | Positive | Slows down |
| Any | Zero | Constant velocity |
Three kinematic equations describe motion with constant acceleration:
To pick one, list what you know and what you want. Look for an equation that contains the unknown you want and values for all its other variables. If none fits, you may need more than one equation.
Graphs carry the same information. On a position-time graph, slope is velocity. Nonzero constant acceleration makes a parabola, while zero acceleration makes a straight line. On a velocity-time graph, slope is acceleration and area is displacement. Area below the time axis counts as negative. On an acceleration-time graph, area is the change in velocity.
Worked example. A car moving at 20 m/s brakes at a constant −5.0 m/s² until it stops. How far does it travel? You know m/s, , and m/s², and you want . Time is missing, so use . Then , so m.
Check with a graph. The stop takes s. The velocity-time graph is a triangle with area m, which matches.
A cart starts from rest and accelerates at 3.0 m/s² for 4.0 s. How far does it go? (Answer: Use m.)
| Graph | Slope gives | Area gives | Constant acceleration looks like |
|---|---|---|---|
| Position vs. time | Velocity | Nothing useful in this course | A parabola for nonzero acceleration, or a straight line for zero acceleration |
| Velocity vs. time | Acceleration | Displacement | A straight line, horizontal if acceleration is zero |
| Acceleration vs. time | Not used here | Change in velocity | A horizontal line |
| Term | Definition |
|---|---|
| Acceleration | The rate of change of velocity, a vector measured in m/s². Average acceleration over an interval is $a_{avg} = \Delta v / \Delta t$. |
| Velocity | The rate of change of position, a vector that includes both speed and direction. |
| Displacement | The change in position, $\Delta x = x_f - x_0$, which can be positive or negative. |
| Constant acceleration | Acceleration that does not change over time, the condition the kinematic equations require. |
| Instantaneous acceleration | The acceleration at one moment, found from the tangent slope on a velocity-time graph. |
| Free fall | Motion under gravity alone. Near Earth's surface, acceleration is about 9.8 m/s² downward for any mass when air resistance is negligible. |
| Acceleration due to gravity (g) | The magnitude of free-fall acceleration near Earth's surface, about 9.8 m/s² (sometimes rounded to 10 m/s²). |
| Velocity-time graph | A graph whose slope gives acceleration and whose signed area gives displacement. Area below the time axis counts as negative. |
| Vector | A quantity with both magnitude and direction, such as displacement, velocity, or acceleration. |
The word "deceleration" makes it feel that way, but the sign only shows direction. Compare signs instead. If velocity and acceleration match, the object speeds up, and if they differ, it slows down.
The ball's velocity is zero for an instant, but its velocity is still changing, so it is still accelerating. Acceleration stays −9.8 m/s² (with up positive) the whole flight, including the top.
A line sloping down on a velocity-time graph means velocity is decreasing. It does not mean the object is moving backward or downhill. Check the axis label first, then ask what the slope and area mean for that graph.
Pick a positive direction at the start and give every vector (displacement, velocity, acceleration) a sign that matches it. The kinematic equations only work for constant acceleration. If acceleration is constant within each stage, you can solve each stage separately. For continuously changing acceleration, a graph can help you find changes in velocity or position from signed areas.