Solving Motion Problems Using Parametric and Vector-Valued Functions
Compare vector displacement and scalar distance, and use velocity and acceleration together for changing speed.
College Board Topic 9.6 · CED p. 171 (PDF p. 176) · CED effective Fall 2020; project course outline 2026–27.
For planar position , velocity is , acceleration is , and speed is . The displacement on [a,b] is . Total distance is . The first is a vector that can cancel; the second is a nonnegative scalar that counts the whole journey.
Average velocity is displacement divided by elapsed time. Average speed is distance divided by elapsed time. A particle can finish where it started and still travel a lot. Use the units: a position vector might be meters, velocity meters per second, and speed also meters per second.
When speed is nonzero, differentiate its square to see whether the particle speeds up: . A positive dot product means increasing speed, a negative one decreasing speed. Do not use the one-dimensional sign shortcut separately on two coordinates and expect it to describe the whole vector.
A derivative table can give exact speed at listed times. A position or velocity table supports numerical estimates of distance or displacement only after a rule and assumptions are stated. Link every number to what is being measured.
Let's work through it
- For on [], compute . Why: Differentiate the position components.
- The speed is , so distance is . Why: Distance accumulates magnitude without cancellation.
- Displacement is . Why: The endpoints agree even though the particle completed a lap.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Worked answer and reason
A magnitude is nonnegative.
Retrieval 2
Worked answer and reason
It is the net vector change in position.
Catch the tempting mistake
- Integrating speed gives the new position.
- Speed loses direction. Integrate velocity componentwise for displacement, then add initial position.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 9 problem 9 · No calculator · verbal
- Unit 9 problem 15 · No calculator · numerical
- Unit 9 problem 16 · No calculator · analytic
- Unit 9 problem 17 · No calculator · verbal
- Unit 9 problem 18 · No calculator · verbal
For the teacher
- Ask for a sentence comparing vector displacement and scalar distance before accepting a motion answer. Delayed practice should mix one- and two-dimensional contexts.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
-
Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
-
Acceleration in calculus
Acceleration is the derivative of velocity with respect to time. In one-dimensional motion, it is also the second derivative of position. For example, s(t)=t² meters gives constant acceleration 2 meters per second squared. A negative acceleration can increase speed when the velocity is also negative.
-
Net change
Net change is the final amount minus the initial amount. Integrating a rate gives this signed change when the model and integral are valid. For example, moving 5 meters right and then 2 meters left gives net displacement 3 meters. Opposite contributions cancel, so net change can differ from the total amount of movement or activity.
-
Vector in calculus
A vector records a magnitude and direction and can be represented by ordered components. In planar motion, the components describe changes or rates along the horizontal and vertical axes. For example, velocity ⟨3,4⟩ has speed 5 because its magnitude is √(3²+4²). A vector-valued function lets those components vary with a parameter.
-
Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.