Displacement, Velocity, and Acceleration

🧭 Plot Summary

Start with the object model: a real object's size, shape, and internal structure get set aside, and it's treated as a single point carrying properties like mass. That point's displacementis just Δx = x − x₀. From there, average velocity (v_avg = Δx/Δt) and average acceleration (a_avg= Δv/Δt) describe an interval as a whole — and an object is accelerating any time its velocity's magnitude or direction changes, even moving at constant speed around a curve. Shrink that interval toward zero, and the average becomes the instantaneous value: v_x = dx/dt and a_x = dv_x/dt. This is where calculus takes over for good — differentiation gets you from position to velocity to acceleration, and integration runs the chain in reverse.

What you'll do in this lesson

• Use the object model to treat a real object as a single point with extensive properties like mass.
• Calculate displacement, Δx = x − x₀, from an object's initial and final positions.
• Calculate average velocity (v_avg = Δx/Δt) and average acceleration (a_avg = Δv/Δt).
• Recognize that an object is accelerating whenever speed or direction changes.
• Understand instantaneous values as the limit of an average as Δt → 0.
• Differentiate position to find instantaneous velocity, and velocity to find instantaneous acceleration.
• Use integration to recover velocity or position functions from acceleration or velocity functions.

Why it matters

This lesson is the calculus hinge of the entire course. Every later unit leans on the idea that instantaneous quantities are derivatives, and that integrating backward recovers the function you started with. Topic 1.3 will put all of this on a graph — but the derivative and integral relationships themselves are set right here.

✅ Self-Check Before You Roll On

Check off each item as you get there. These aren't grades — they're your own signal.