Kinematics Formulas

Why This Matters

Kinematics is the foundation of everything you'll study in AP Physics 1—it's the language of motion. Before you can analyze forces, energy, or momentum, you need to describe how objects move: where they are, how fast they're going, and how their motion changes. The formulas in this guide aren't just equations to memorize; they represent fundamental relationships between position, velocity, acceleration, and time that you'll apply throughout the entire course.

You're being tested on your ability to choose the right equation for a given scenario and interpret what each variable means physically. The AP exam loves problems where you must recognize which quantity is missing (often time!) and select the appropriate formula. Don't just memorize these equations—understand when each one applies, what assumptions it requires (usually constant acceleration), and how the variables connect. That conceptual understanding is what separates a 3 from a 5.

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Defining Motion: The Foundational Relationships

These equations define what velocity and acceleration mean. They work for any motion—constant, accelerating, or irregular—because they describe averages over intervals.

Average quantities tell you the overall behavior between two points, regardless of what happens in between.

Average Velocity (Definition)

• $v_{avg} = \frac{\Delta x}{\Delta t}$—this is the definition of average velocity, not a derived formula
• Applies to any motion, whether acceleration is constant, changing, or zero
• Displacement matters, not distance—if you return to your starting point, average velocity is zero even if you traveled far

Average Acceleration (Definition)

• $a = \frac{\Delta v}{\Delta t}$—defines acceleration as the rate of change of velocity
• Sign indicates direction, not speeding up vs. slowing down (negative acceleration can mean speeding up in the negative direction)
• Units are $m/s^2$, which you can read as "meters per second, per second"

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The Big Three: Constant Acceleration Equations

These are the workhorses of kinematics problems. Each equation connects four of the five kinematic variables ($x$, $v_0$, $v$, $a$, $t$), leaving one out. Your job is to identify which variable is missing and pick the equation that doesn't need it.

All three equations assume constant (uniform) acceleration—they don't work if acceleration is changing.

Velocity-Time Equation

• $v = v_0 + at$—directly shows how velocity changes over time due to constant acceleration
• Missing variable: displacement ($\Delta x$)—use this when you don't know or need position
• Graphically, this equation represents a straight line on a $v$-$t$ graph with slope $a$ and intercept $v_0$

Position-Time Equation

• $\Delta x = v_0 t + \frac{1}{2}at^2$—calculates displacement when you know initial velocity, acceleration, and time
• Missing variable: final velocity ($v$)—perfect for "how far does it travel in the first 3 seconds?" problems
• The $\frac{1}{2}at^2$ term accounts for the changing velocity; without it, you'd only get the distance at constant speed

Velocity-Displacement Equation

• $v^2 = v_0^2 + 2a\Delta x$—connects velocities to displacement without involving time
• Missing variable: time ($t$)—the go-to equation when time isn't given or asked for
• Relates to energy concepts—this equation foreshadows the work-energy theorem you'll study later ($\frac{1}{2}mv^2$ and $Fd$)

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Special Case: Constant Velocity Motion

When acceleration equals zero, the kinematic equations simplify dramatically. Constant velocity means the object covers equal displacements in equal time intervals.

Position with Constant Velocity

• $x = x_0 + vt$—position changes linearly with time when there's no acceleration
• This is just $\Delta x = v_0 t + \frac{1}{2}at^2$ with $a = 0$—not a separate equation to memorize
• Graphically, this produces a straight line on a position-time graph with slope equal to velocity

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Average Velocity Shortcuts

These formulas provide efficient ways to calculate displacement when you know velocities but want to avoid the full kinematic equations. They're derived from the Big Three, so they only work under constant acceleration.

Arithmetic Mean Method

• $v_{avg} = \frac{v + v_0}{2}$—average velocity equals the midpoint between initial and final velocities
• Only valid for constant acceleration—if acceleration varies, this shortcut fails
• Combine with $\Delta x = v_{avg} \cdot t$ to find displacement quickly when you know both velocities and time

Displacement from Average Velocity

• $\Delta x = v_{avg} \cdot t$—displacement equals average velocity multiplied by time
• Equivalent to $\Delta x = \frac{1}{2}(v + v_0)t$ when you substitute the arithmetic mean formula
• Useful for mental math—sometimes faster than plugging into $\Delta x = v_0 t + \frac{1}{2}at^2$

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Vertical Motion: Free Fall Applications

Free fall is just a special case of constant acceleration where $a = g$ (gravitational acceleration). The same kinematic equations apply—you're just substituting $g$ for $a$ and using vertical variables.

Vertical Displacement Equation

• $\Delta y = v_{0y}t + \frac{1}{2}gt^2$—identical structure to $\Delta x = v_0 t + \frac{1}{2}at^2$, with $g \approx 9.8 \, m/s^2$ or $10 \, m/s^2$
• Sign convention matters—choose positive direction (usually up) and be consistent; if up is positive, $g = -9.8 \, m/s^2$
• Foundation for projectile motion—vertical and horizontal components are analyzed independently using these equations

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Quick Reference Table

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Self-Check Questions

1. Which kinematic equation would you use to find how far a car travels if you know its initial velocity, final velocity, and acceleration—but not time?

2. Compare $v_{avg} = \frac{\Delta x}{\Delta t}$ and $v_{avg} = \frac{v + v_0}{2}$: Under what conditions does the second formula apply, and when might it give a wrong answer?

3. An object is thrown upward and returns to its starting height. Using $v^2 = v_0^2 + 2a\Delta x$, what is its final speed compared to its initial speed? Why?

4. If you're given a position-time graph that's curved (parabolic), which kinematic equation best describes the motion, and what does the curvature tell you about acceleration?

5. In a projectile motion FRQ, you need to find the time an object is in the air. Which equation would you apply to the vertical component, and why can't you use the horizontal component to find time directly?