In AP Calculus, speed is the magnitude of velocity: |v(t)| for straight-line motion, or √((dx/dt)² + (dy/dt)²) for planar motion (BC). It measures how fast a particle moves with no direction attached, and integrating speed gives total distance traveled.
Speed is velocity with the direction stripped off. For a particle moving along a line with velocity v(t), speed is |v(t)|. Velocity can be negative (the particle is moving left or down), but speed never can. A particle with v(t) = -7 has a speed of 7.
In BC's Unit 9, the same idea goes 2D. A particle moving in the plane has a velocity vector ⟨x'(t), y'(t)⟩, and its speed is the magnitude of that vector, √((x'(t))² + (y'(t))²). Either way, speed answers one question: how fast, right now? And it comes with a bonus rule worth memorizing. The definite integral of speed over a time interval gives total distance traveled, while the integral of velocity only gives displacement (net change in position).
Speed lives in Topic 4.2 (Straight-Line Motion) under learning objective 4.2.A, where the derivative connects position, velocity, speed, and acceleration in rectilinear motion problems. It returns in Topic 9.6 for BC under learning objective 9.6.A, where derivatives of parametric and vector-valued functions give velocity, speed, and acceleration for planar motion (FUN-8.B.1), and the definite integral of speed gives total distance traveled (FUN-8.B.2). Motion problems are one of the most reliable FRQ setups on both the AB and BC exams, and almost every one of them tests whether you know that speed and velocity are not the same thing. The classic trap question, "is the particle speeding up?", can't be answered without understanding speed as |v(t)|.
Keep studying AP Calculus Unit 4
Visual cheatsheet
view galleryAverage Velocity (Unit 4)
Velocity is the signed version of speed. Velocity tells you how fast AND which way; speed throws away the sign. A particle can have negative velocity and increasing speed at the same time, which is exactly the trap the exam loves.
Speeding up (Unit 4)
A particle speeds up when velocity and acceleration have the same sign, and slows down when they have opposite signs. This works because speed is |v(t)|, so making v more negative still makes the particle faster. Sign charts of v(t) and a(t) settle it instantly.
Distance Traveled (Units 6 and 9)
Integrate velocity and you get displacement, where backward motion cancels forward motion. Integrate speed and nothing cancels, so you get total distance traveled. One absolute value sign is the entire difference between the two answers.
Acceleration Vector (Unit 9)
In planar motion, speed is the length of the velocity vector ⟨x'(t), y'(t)⟩. The acceleration vector ⟨x''(t), y''(t)⟩ tells you how that velocity vector is changing. BC FRQs routinely ask for the speed at a specific time, which is just plugging into the magnitude formula.
Motion FRQs show up almost every year, and speed is baked into them. The 2017 FRQ (Q5) and 2021 FRQ (Q2) both gave particles on the x-axis and asked questions that hinge on the velocity-versus-speed distinction, and the 2023 BC FRQ Q2 used a particle on a curve with parametric components, where speed is the magnitude of the velocity vector. Multiple-choice stems typically hand you a position function like x(t) = 2t² - 6t + 5 or a velocity function like v(t) = 4 - t and ask which quantity is negative at a given time, or during which interval the particle is speeding up or slowing down. To earn the points, you need to do three things on command: compute |v(t)| or √((x')² + (y')²) at a specific time, compare the signs of v and a to decide speeding up versus slowing down, and set up ∫|v(t)|dt when the question says "total distance" instead of "displacement."
Velocity is a signed (or vector) quantity that includes direction; speed is its absolute value or magnitude. So v(t) = -3 means velocity is negative but speed is 3. This matters most for "speeding up or slowing down" questions. The particle speeds up whenever v and a share a sign, even if both are negative, because speed |v| is growing. Saying "velocity is increasing" when you mean "speed is increasing" will cost you points on an FRQ justification.
Speed is the absolute value of velocity, |v(t)|, so it is never negative even when the particle moves in the negative direction.
For planar motion (BC, Topic 9.6), speed is the magnitude of the velocity vector: √((x'(t))² + (y'(t))²).
A particle is speeding up when velocity and acceleration have the same sign, and slowing down when they have opposite signs.
The definite integral of speed gives total distance traveled, while the definite integral of velocity gives displacement.
On FRQs, always justify speed answers by referring to |v(t)| or the signs of v and a, not just velocity alone.
Speed is the magnitude of velocity: |v(t)| for motion along a line (Topic 4.2) or √((x'(t))² + (y'(t))²) for motion in the plane (Topic 9.6, BC only). It measures how fast a particle moves without any direction information.
No. Velocity carries a sign or direction, while speed is always nonnegative. A particle with v(t) = -5 is moving in the negative direction at a speed of 5, and confusing the two is one of the most common point-losers on motion FRQs.
Compare the signs of velocity v(t) and acceleration a(t). Same sign means speeding up; opposite signs mean slowing down. For example, with v(t) = 4 - t and a(t) = -1, the particle slows down on (0, 4) and speeds up for t > 4.
Integrating speed, ∫|v(t)|dt, gives total distance traveled. Integrating velocity, ∫v(t)dt, gives displacement (net change in position), where backward motion cancels forward motion. The CED states this directly in FUN-8.B.2.
Take the magnitude of the velocity vector. If position is (x(t), y(t)), then speed = √((dx/dt)² + (dy/dt)²). The 2023 BC FRQ Q2 used exactly this setup with a particle moving along a curve.