Physics · Unit 1: Kinematics · Activity 1-2-2

Deep Dive: The Kinematic Equations

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

1.2.2.AConcept

The Kinematic Toolkit

Every constant-acceleration motion problem involves the same five quantities: displacement (Δx), initial velocity (v₀), final velocity (v), acceleration (a), and time (t). The four kinematic equations are just four different ways of relating these five variables — and each equation leaves exactly one of the five out.

🔑These equations only work for constant acceleration. If acceleration is changing during the motion you're analyzing, none of the four apply — you'd need calculus-based tools this course doesn't cover. Always ask whether constant acceleration is a reasonable assumption before reaching for any of them.

1.2.2.BMath

Deriving v = v₀ + at

This one is just Lesson 1-2-1's slope formula, rearranged. Acceleration is the slope of a v-t graph:

a = Δv / Δt = (v − v₀) / t

Multiply both sides by t, then add v₀ to both sides:

v = v₀ + at

This equation leaves out Δx entirely — notice it never mentions displacement. That makes it the natural choice whenever a problem never gives or asks for how far something traveled.

1.2.2.CMath

Deriving Δx = v₀t + ½at²

This is the rectangle-plus-triangle area split from Lesson 1-2-1, written as algebra instead of geometry. Picture a v-t graph with constant acceleration: velocity starts at v₀ and rises steadily to v over time t.

💡Split the area under that line into a rectangle (base t, height v₀) sitting below a triangle (base t, height v − v₀ = at, since v = v₀ + at).

rectangle area = v₀t   |   triangle area = ½ · t · at = ½at²

Add the two pieces together:

Δx = v₀t + ½at²

This equation leaves out v (the final velocity) — everything it needs comes from the starting velocity, the acceleration, and the time.

ExampleWorked Example — Finding Displacement Without Final Velocity

A skateboarder starts at 2 m/s and accelerates at 1.5 m/s² for 4 seconds. Find her displacement during this time.

1.2.2.DMath

Deriving Δx = ½(v₀ + v)t

You've actually already derived this one — it's the exact trapezoid-area worked example from Lesson 1-2-1's Deep Dive. Averaging the starting and ending velocity gives the trapezoid's "effective height":

average velocity = (v₀ + v) / 2

Multiply that average by the total time to get the trapezoid's area:

Δx = ½(v₀ + v)t

This equation leaves out a — acceleration never appears, because you don't actually need it if you already know both velocities and the time.

🔑A fifth relationship, v² = v₀² + 2aΔx, comes from combining the other three algebraically (substitute t from one equation into another and simplify). It leaves out t entirely, which makes it the go-to choice whenever a problem never mentions how long something took.

1.2.2.EConcept

The Missing-Variable Strategy

Instead of memorizing "problem types," get in the habit of listing the five variables and marking each one Given, Wanted, or Not Mentioned. The one Not Mentioned variable is the one whose equation you should use — with one exception.

⚠️v₀ shows up in every one of the four equations, so it can never be the "Not Mentioned" variable. If a problem seems to skip v₀ entirely, look for a disguised value first — "starts from rest" means v₀ = 0, it isn't missing information.

Mark which three variables a problem GIVES you, then pick which one you want to FIND. The variable left over — never given, never asked for — is the one that tells you which equation to use.

Δxdisplacement

v₀initial velocity

vfinal velocity

aacceleration

ttime

Δx (displacement) is never given and never asked for — use:

v = v₀ + at

⚠️A common mistake: picking an equation because it "looks like" the one from a similar homework problem, instead of checking which variable is actually missing. Two problems can look similar on the surface and require completely different equations once you check what's actually given.

1.2.2.FExample

Multi-Step Problems

Some problems can't be solved with a single equation because you don't yet have three of the five variables for the quantity you actually want. The fix: solve for an intermediate unknown first, then use it as a known in a second equation.

ExampleGuided Example — A Two-Stage Problem

A cart starts at rest and accelerates at 2 m/s² for 5 seconds, then continues at whatever velocity it reached for another 6 seconds at constant velocity. Find the cart's total displacement over both stages.

Step 1 — Identify what's missing in Stage 1

Stage 1: v₀ = 0, a = 2 m/s², t = 5 s. Target: v (needed for Stage 2). Δx is not needed yet, so use v = v₀ + at.

← Back to Activity 1-2-2 Do the Activity →Next up: Project 1-2-3, Reconstruct the Ride.