Every kinematic equation assumes the acceleration is constant.
Acceleration The rate of change of velocity with time.
Step 1: Something to Notice
Watch first. The explanation comes later.
A stone dropped from a bridge falls 5 m in the first second and 15 m in the second — three times as far in the same time.
Why is the second interval not the same as the first?
Step 2: Find Out
Isolate the kind of motion and note the shape each graph takes.
Step 1 — Predict
What do all four kinematic equations assume?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
v = v₀ + at comes straight from the definition
Constant rate of change over a time interval.
Average velocity is the mean of the two ends
True only because the velocity–time graph is a straight line.
v² = v₀² + 2aΔx eliminates time
The one to reach for when no time is given or wanted.
Pick a sign convention and keep it
At the top of a throw, velocity is zero and acceleration is not.
Four equations, one assumption
The kinematic equations assume constant acceleration. Applying them to a situation where acceleration varies gives a confident wrong answer, and checking that assumption is the first step.
Choose by what is missing
Each equation omits one of the five quantities. Listing what you know and what you want identifies which equation to use without trial and error.
Step 4: A Common Mistake
Lots of people think
“The kinematic equations apply to any motion.”
Step 5: Where the Equations Come From If You Know Calculus
The same idea somewhere new.
Each kinematic equation is an integral. Acceleration is dv/dt, so integrating a constant acceleration once gives v = v₀ + at, and integrating again gives x = x₀ + v₀t + ½at². The constants of integration are the initial velocity and the initial position, which is why they appear where they do rather than by convention. Seeing it this way also explains the limitation exactly: the integration is trivial only because a is constant and comes outside the integral. For non-constant acceleration the same integrals still apply and simply require a different function inside — which is what a physics course with calculus does next, and why this chapter is called calculus readiness.
Step 6: Think It Through
Practice makes it stick.
At the Top of a Throw
Problem 1 of 2
A ball is thrown straight up. At the highest point, what is its acceleration?
A Falling Coffee Filter
Problem 2 of 2
Can the kinematic equations give its fall time?
Four Equations
1 of 5
What do the kinematic equations assume?
2 of 5
Which equation has no time in it?
3 of 5
Why is average velocity the mean of the two ends?
4 of 5
At the top of a throw, what is zero?
5 of 5
How do you choose which equation to use?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Do the kinematic equations work for any motion?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Derive the kinematic equations and state exactly what they assume.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- All four kinematic equations assume constant acceleration.
- They follow from the definitions of velocity and acceleration, plus a straight-line v–t graph.
- v² = v₀² + 2aΔx is the one with no time in it.
- Choose a sign convention at the start and keep it throughout.