Vector-Valued Functions & Motion
- Differentiate a vector-valued position function to get velocity and acceleration
- Compute the speed of a particle moving in the plane
- Find the position from velocity using integration and an initial condition
Motion in the plane as a vector
A particle moving in the plane has a position vector ⟨x(t), y(t)⟩. Differentiating component-by-component gives the velocity vector ⟨x′(t), y′(t)⟩, and differentiating again gives the acceleration vector ⟨x″(t), y″(t)⟩. Each component is handled with ordinary single-variable calculus; the vector just bundles the two independent motions together. This is exactly the parametric setup, now read as physical motion.
Speed is the magnitude of velocity
The velocity vector ⟨x′, y′⟩ points in the direction of motion, but speed is its scalar length √((x′)² + (y′)²). This is the same Pythagorean combination that appears in arc length — no accident, since speed is the rate at which arc length accumulates. To find total distance traveled over [a, b], integrate the speed: ∫ₐᵇ √((x′)² + (y′)²) dt. Speed is never negative, while a velocity component can be.
Recovering position from velocity
Given the velocity vector and a starting position, you recover position by integrating each component and using the initial condition to fix each constant. x(t) = x(t₀) + ∫_{t₀}^t x′(s) ds, and likewise for y. On the calculator section this is often done as a definite integral added to the known starting coordinate — the Fundamental Theorem of Calculus applied one component at a time.
A particle has velocity vector v(t) = ⟨2t, 3⟩. Find its speed at t = 2 and its acceleration vector.
- 1.Speed is the magnitude of velocity: |v| = √((2t)² + 3²) = √(4t² + 9).
- 2.At t = 2: √(4·4 + 9) = √(16 + 9) = √25 = 5.
- 3.Acceleration is the derivative of velocity, component by component: a(t) = ⟨d/dt[2t], d/dt[3]⟩.
- 4.This gives a(t) = ⟨2, 0⟩, constant for all t.
Speed is a single nonnegative number — the magnitude √((x′)² + (y′)²) — not a vector. Reporting speed as ⟨x′, y′⟩ confuses it with velocity. Whenever a problem asks "how fast," it wants the scalar magnitude.
A particle has velocity vector ⟨3, 4⟩ at a certain instant. What is its speed at that instant?
If a particle’s position is ⟨t², sin(t)⟩, what is its acceleration vector?
For planar motion, keep three vectors straight: position ⟨x, y⟩, velocity ⟨x′, y′⟩, acceleration ⟨x″, y″⟩. "Speed" and "total distance" both use the magnitude √((x′)² + (y′)²) — speed at an instant, distance as its integral over time.
Answer the 2 checkpoints as you read.
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