Kinematics Equations

Why This Matters

Kinematics is the foundation of everything you'll study in mechanics—and mechanics typically makes up the largest portion of your introductory physics exams. These equations describe how objects move through space and time, connecting displacement, velocity, acceleration, and time in predictable ways. Master these relationships, and you'll have the tools to analyze everything from a ball thrown in the air to a car braking at a stoplight.

Here's what you're really being tested on: not just plugging numbers into formulas, but choosing the right equation for each problem. Every kinematics equation leaves out one variable—and that's your key to solving problems efficiently. You'll also need to understand when these equations apply (constant acceleration only) and how to interpret their physical meaning. Don't just memorize the formulas—know what each one tells you about motion and when to reach for it.

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The Foundational Definitions

Before diving into the "big five" kinematics equations, you need to understand the definitions that underpin them. These equations define what velocity and acceleration actually mean and apply to all motion scenarios.

Velocity Definition

• $v = \frac{\Delta x}{t}$—defines velocity as displacement divided by time for constant-velocity motion
• Only valid when acceleration is zero; this is your starting point for uniform motion problems
• Rearranges to $\Delta x = vt$, which you'll use constantly for objects moving at steady speed

Acceleration Definition

• $a = \frac{\Delta v}{t} = \frac{v - v_0}{t}$—defines acceleration as the rate of change of velocity
• Positive acceleration means speeding up in the positive direction (or slowing down in the negative direction)
• This definition rearranges into the first kinematic equation, making it foundational for all accelerated motion

Average Velocity

• $\bar{v} = \frac{v + v_0}{2}$—average velocity equals the mean of initial and final velocities
• Only valid for constant acceleration; non-uniform acceleration requires calculus
• Connects to displacement through $\Delta x = \bar{v} \cdot t$, giving you an alternative problem-solving path

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The Big Five Kinematic Equations

These are your primary problem-solving tools for constant acceleration. Each equation connects four of the five kinematic variables ($x$, $v$, $v_0$, $a$, $t$), leaving one out—and that's how you choose which equation to use.

Velocity-Time Equation (Missing $\Delta x$)

• $v = v_0 + at$—directly derived from the definition of acceleration, rearranged
• Use when displacement isn't given or needed; perfect for "how fast is it going after 5 seconds?" problems
• Graphically represents the equation of a line on a velocity-time graph, where $a$ is the slope and $v_0$ is the y-intercept

Position-Time Equation (Missing $v$)

• $\Delta x = v_0 t + \frac{1}{2}at^2$—calculates displacement without needing final velocity
• The $\frac{1}{2}at^2$ term represents additional displacement due to acceleration beyond what initial velocity provides
• Most common equation for projectile problems where you know initial conditions and time, but not final speed

Velocity-Displacement Equation (Missing $t$)

• $v^2 = v_0^2 + 2a\Delta x$—the time-independent equation, connecting velocities to displacement
• Use when time isn't given; essential for problems like "how fast is the car going after traveling 100 meters?"
• Derives from energy concepts—this equation is actually a disguised work-energy relationship

Average Velocity Displacement (Missing $a$)

• $\Delta x = \frac{1}{2}(v + v_0)t$—calculates displacement using average velocity
• Use when acceleration isn't given but you know both initial and final velocities
• Equivalent to finding the area of a trapezoid under a velocity-time graph

Full Position Equation (Missing nothing—includes $x_0$)

• $x = x_0 + v_0 t + \frac{1}{2}at^2$—gives final position, not just displacement
• Use when starting position isn't zero; essential for problems with defined coordinate systems
• The most complete equation for tracking an object's actual location in space

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Quick Reference Table

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Self-Check Questions

1. An object accelerates uniformly from rest. Which equation would you use to find its displacement after 4 seconds if you don't know its final velocity?

2. Compare $v = v_0 + at$ and $v^2 = v_0^2 + 2a\Delta x$: both can find final velocity, so what determines which one you should use?

3. A car brakes to a stop over a known distance. You want to find how long the braking took. Which variable is "missing" from your known quantities, and which equation should you use?

4. Why does the average velocity equation $\bar{v} = \frac{v + v_0}{2}$ only work for constant acceleration? What would change if acceleration varied?

5. An FRQ describes a ball thrown upward and asks for maximum height. Which kinematic equation is most efficient here, and what value do you substitute for $v$?