Physics 1 (Calculus-Based) / Unit 101

What you'll learn

• Differentiate a position function to obtain velocity and acceleration functions, and evaluate them at a specific time
• Integrate an acceleration function, using an initial condition to solve for the constant of integration, to recover velocity and position functions
• Solve motion problems with non-constant (time-dependent) acceleration, which algebra-based kinematics has no tools for
• Derive the constant-acceleration kinematic equations as a special case of integrating a constant a(t), rather than treating them as separate memorized formulas
• Distinguish average velocity (a displacement-over-time ratio, or equivalently a time-average of v(t)) from instantaneous velocity (dx/dt at one instant), and explain why they generally differ
• Determine whether an object is speeding up or slowing down from the signs of v(t) and a(t), and locate turning points where v(t) = 0

1. Position, Velocity, and Acceleration as Derivatives

In an algebra-based course, velocity and acceleration show up as separate quantities connected by a handful of formulas that only work for constant acceleration. Here, they're related by one operation, applied twice: velocity is the derivative of position, and acceleration is the derivative of velocity — \(v(t) = \dfrac{dx}{dt}\), \(a(t) = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}\). Given any position function, however complicated, differentiating it once gives velocity and differentiating again gives acceleration.

The sign of v(t) gives direction (positive or negative along whatever axis x is measured on), and its magnitude gives speed. A particle is at rest — momentarily or permanently — exactly when v(t)=0; whether that's a brief turning point or a lasting stop depends on what v(t) does on either side of that instant, which is itself a question you answer by looking at the function, not by memorizing a rule.

2. Recovering Motion from Acceleration: Integration and Initial Conditions

Differentiation goes from position to velocity to acceleration. Going the other way — from acceleration back to velocity, or velocity back to position — means integrating, and every indefinite integral leaves behind an unknown constant. That constant is exactly what an initial condition (a given value of \(v\) or \(x\) at some specific time, usually \(t=0\)) is for: it pins down the one antiderivative that actually matches the particle's real history, out of the whole family that differentiation alone can't distinguish between.

This is also where the "memorized" constant-acceleration formulas from an algebra-based course actually come from. They aren't a separate set of rules — they're what you get from integrating a constant \(a(t)=a_0\) exactly once (to get \(v(t) = v_0 + a_0t\)) and twice (to get \(x(t) = x_0 + v_0t + \tfrac{1}{2}a_0t^2\)). Once you can integrate, those formulas stop being things to memorize and start being a special case you could rederive in ten seconds.

3. Motion Under Non-Constant Acceleration

An algebra-based course's kinematic equations only work when acceleration is constant — the moment \(a\) depends on time, those formulas simply don't apply, and there's no algebra-only way to patch them. This is the actual reason calculus shows up in mechanics at all: \(a(t) = \dfrac{dv}{dt}\) and its integral form work identically whether \(a\) is constant or not, so a time-varying acceleration is no harder to handle in principle — the integral just isn't a straight multiplication anymore.

4. Average vs. Instantaneous Quantities

Instantaneous velocity, \(v(t)=dx/dt\), describes motion at one exact instant. Average velocity over an interval is a completely different computation — the net displacement divided by the elapsed time, \(v_{avg} = \Delta x/\Delta t\) — and in general there's no reason to expect these to agree at either endpoint of the interval, or at its midpoint. The Mean Value Theorem guarantees some instant inside the interval where they're equal, but "some instant" isn't necessarily anywhere obvious.

Key equations

• v(t) = dx/dt — Velocity is the instantaneous rate of change of position — differentiate x(t) to get v(t).
• a(t) = dv/dt = d²x/dt² — Acceleration is the instantaneous rate of change of velocity (the second derivative of position).
• v(t) = v₀ + ∫₀ᵗ a(τ) dτ — Recovering velocity from acceleration — works for constant OR time-varying a(t), unlike any single algebra-based formula.
• x(t) = x₀ + ∫₀ᵗ v(τ) dτ — Recovering position from velocity, by the same integration process.
• v_avg = Δx/Δt = (1/Δt)∫v(t) dt — Average velocity over an interval — a displacement/time ratio, not a value of v(t) at any particular instant.

Open interactive practice for this unit