9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

Position, Velocity, and Acceleration Vectors

For a particle with position x(t), y(t), the velocity vector is ⟨x'(t), y'(t)⟩ and acceleration is ⟨x''(t), y''(t)⟩. These vectors describe motion in the plane, extending single-variable calculus ideas: derivatives give rates of change of each coordinate with respect to time.

Speed and Distance Traveled

Speed is the magnitude of velocity: √[(x'(t))² + (y'(t))²]. Total distance traveled along the path from t = a to t = b is ∫ₐᵇ √[(x'(t))² + (y'(t))²] dt — the same formula as arc length. This differs from displacement, which only measures net change in position.

Finding Position from Velocity

If given velocity components and an initial position, use the Fundamental Theorem of Calculus: x(t) = x(t₀) + ∫ₜ₀ᵗ x'(s) ds, and similarly for y(t). This lets you reconstruct the full position vector when only velocity and a starting point are known.

Key terms

Velocity vector

⟨x'(t), y'(t)⟩ — the rate of change of position with respect to time.

Acceleration vector

⟨x''(t), y''(t)⟩ — the rate of change of velocity with respect to time.

Speed formula

√[(x'(t))² + (y'(t))²] — the magnitude of the velocity vector.

Total distance traveled

∫ₐᵇ √[(x'(t))² + (y'(t))²] dt — the arc length of the path from t = a to t = b.

Finding position from velocity

Use the Fundamental Theorem of Calculus: x(t) = x(t₀) + ∫ₜ₀ᵗ x'(s) ds, and similarly for y(t).

Try an AP-style question

r(t) = ⟨t², t³⟩. What is the particle's speed at t = 1?

1. 5
2. √13
3. 13
4. √5