3.2 Representing Acceleration with Equations and Graphs

Kinematic equations describe how objects move under constant acceleration, connecting velocity, position, and time. Mastering these equations and their graphical representations is essential for solving nearly every problem you'll encounter in this unit and beyond.

Kinematics Equations and Graphs

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Interpretation of kinematic equations

Three core equations govern motion with constant acceleration. Each one links a different combination of variables, and the key to using them well is knowing which variable each equation leaves out.

• $v = v_0 + at$ connects final velocity ($v$), initial velocity ($v_0$), acceleration ($a$), and time ($t$). It tells you that velocity changes linearly with time when acceleration is constant. This equation has no position term.
• $x = x_0 + v_0 t + \frac{1}{2}at^2$ connects position ($x$), initial position ($x_0$), initial velocity ($v_0$), acceleration ($a$), and time ($t$). The $\frac{1}{2}at^2$ term is what makes position change non-linearly when an object accelerates. This equation has no final velocity term.
• $v^2 = v_0^2 + 2a(x - x_0)$ connects final velocity ($v$), initial velocity ($v_0$), acceleration ($a$), and displacement ($x - x_0$). This is your go-to when you don't know time and aren't asked to find it. This equation has no time term.

All three equations assume constant acceleration and neglect air resistance. They apply directly to free-fall problems near Earth's surface, where $a = g \approx 9.8 \text{ m/s}^2$ downward.

Problem-solving with constant acceleration

1. List your knowns and your unknown. Write down every given quantity with its sign and units, and identify the single variable you need to find.
2. Pick the right equation. Choose the kinematic equation that contains your unknown and uses only your known variables. A quick way to decide: figure out which variable is missing from the problem entirely (not given and not asked for), then use the equation that also leaves out that variable.
3. Substitute known values into the equation, keeping signs and units consistent.
4. Solve algebraically for the unknown. Isolate it on one side before plugging in numbers whenever possible; this reduces arithmetic errors.
5. Check your answer. Verify that the units work out and that the magnitude and sign make physical sense. A car accelerating from rest for 5 seconds shouldn't end up traveling 10 km, for instance.

Relationships in kinematic graphs

Each type of graph reveals different information about an object's motion, and the graphs are all connected to one another.

Position vs. time graph

• The slope of the tangent line at any point gives the instantaneous velocity at that moment. A steeper slope means a faster object.
• Under constant acceleration, the curve is a parabola. It's concave up when acceleration is positive and concave down when acceleration is negative.

Velocity vs. time graph

• The slope of the line equals the acceleration. A straight line with positive slope means constant positive acceleration; a negative slope means constant negative acceleration.
• The area between the curve and the time axis equals the displacement over that interval. Area above the axis counts as positive displacement, and area below counts as negative.

Acceleration vs. time graph

• Constant acceleration appears as a horizontal line. Its value (positive or negative) tells you the direction of the acceleration.
• The area under the curve equals the change in velocity over that time interval.

How the graphs connect to each other

If you can read slopes and areas on one graph, you can reconstruct the other two. Practice moving between all three representations; exam questions frequently ask you to do exactly that.

Vector and Scalar Quantities in Kinematics

• Vector quantities have both magnitude and direction. In kinematics, velocity, acceleration, and displacement are all vectors. Their signs in 1-D problems indicate direction (e.g., positive for right/up, negative for left/down).
• Scalar quantities have magnitude only. Speed, time, and distance are scalars; they're always zero or positive.
• Motion diagrams represent vectors visually at equal time intervals. The spacing between dots shows how fast the object moves, and arrows indicate the direction and relative size of velocity or acceleration at each instant.