Covers: Motion with derivatives and integrals, vectors and projectiles. Quiz one topic at a time or take the whole unit.
The idea each topic turns on, and the mistake students make most often.
Key ideaVelocity is dx/dt and acceleration is dv/dt. Differentiate the function, then substitute the time.
Common mistakeSubstituting the time before differentiating, or dividing x by t (an average) when the question asks for an instantaneous value.
Key ideaIntegrate acceleration to get the change in velocity and velocity to get the change in position, then add the initial value: x(t) = x₀ + ∫v dt.
Common mistakeLeaving out the constant of integration (the initial velocity or position).
Key ideaSlopes take you down the chain (x → v → a) and areas take you up (a → Δv → Δx).
Common mistakeReading values off the wrong graph, or treating an area below the axis as positive.
Key ideaWhen the acceleration changes, use calculus (or v dv/dx = a when a depends on x). The constant-acceleration equations don’t apply.
Common mistakePlugging a time-varying acceleration into v = v₀ + at.
Key ideaThe velocity of A relative to C is the velocity of A relative to B plus the velocity of B relative to C, added as vectors.
Common mistakeSubtracting in the wrong order, or adding magnitudes of velocities that are at an angle.
Key ideaTreat x and y independently, with separate component equations that share the same t.
Common mistakePutting the total speed into a single-axis equation (v instead of vy in the vertical equation), or giving a projectile a horizontal acceleration.
This is one of the 30 questions in the Unit 1 quiz.
A particle’s position is , with in meters and in seconds. What is its velocity at s?
Velocity is the derivative of position. Differentiate, then evaluate at s.
Answer: A. 10 m/s
Velocity is the derivative of position: , so m/s.
AP-style FRQs for this unit. Students score themselves point by point against a rubric and see a full solution.
