PHYS2150 Chap.5 Vector Motion, Arc Length and Curvature
Vector Motion, Arc Length and Curvature
A vector function carries a path and a clock
A space curve may be represented by r(t)=f(t)i+g(t)j+h(t)k. Its component domains must all be valid, and increasing t chooses an orientation. Different parametrisations can trace the same geometric path at different rates. Limits, continuity, derivatives and integrals are taken componentwise.
A smooth curve has a continuous, nonzero derivative; when r′ vanishes, the usual tangent and normal formulas may fail even if the traced set itself looks continuous.
Velocity and acceleration answer different questions
When t is time, r is position, v=r′ is velocity, and a=v′=r″ is acceleration. Velocity is tangent to the path and includes direction; speed is the scalar ||v||.
Acceleration measures change of the full velocity vector, so it can be nonzero even when speed is constant. If acceleration and initial data are given, integrate componentwise, apply the velocity condition after the first integration, and apply the position condition after the second.
The constants of integration are vectors.
Arc length removes dependence on traversal rate
The length from t=a to t=b is L=∫||r′(t)||dt. The corresponding arc-length function s(t)=∫ from a to t of ||r′(u)||du satisfies ds/dt=||r′(t)||. Reparametrising by s makes the derivative a unit vector, separating the geometry of the curve from the rate at which a particle travels along it.
This also provides a direct check: an arc-length parametrisation must have derivative of magnitude one.
The moving frame records the bend
The unit tangent is T=v/||v||. Where T changes, the principal unit normal is N=T′/||T′|| and points toward the concave side. Curvature is κ=||dT/ds||. With a general parameter, κ=||T′(t)||/||r′(t)||; in three dimensions the equivalent formula κ=||v×a||/||v||³ is often faster.
The radius of curvature is 1/κ, and the centre of the osculating circle is r+(1/κ)N. A centre on the convex side reveals a sign or normal-direction error.
Acceleration splits into tangential and normal parts
Writing a=aT T+aN N separates two physical effects. The tangential magnitude aT=dv/dt changes speed, while the normal magnitude aN=κv² changes direction.
These components are perpendicular, so the total magnitude is found by a Pythagorean combination rather than ordinary addition.
Uniform circular motion has aT=0 but inward aN≠0. Straight-line motion has κ=0 and therefore no normal acceleration, although its speed may still change.
Central forces connect calculus to orbital geometry
For inverse-square gravity, acceleration is parallel and opposite to r, so r×a=0. Therefore d(r×v)/dt=0 and r×v is a constant vector.
The orbit lies in a fixed plane perpendicular to that vector, and one half of ||r×v|| is the constant areal velocity, explaining equal areas in equal times. Continuing the inverse-square analysis produces a conic orbit r=p/[1+e cos(θ−θ₀)]; bound planetary motion has 0≤e<1 and places the Sun at a focus.
What this chapter covers
- 01
Vector functions and derivatives
- 02
Velocity, speed and acceleration
- 03
Arc length parametrisation
- 04
Tangent and normal vectors
- 05
Curvature and osculating circles
- 06
Tangential and normal acceleration
- 07
Central forces and Keplerian motion
Worked example · free
Curvature of a helix
- 2Differentiate position twice.
- 4Evaluate speed and cross product.
- 2Interpret curvature.
Key terms
- Unit Tangent
- The velocity vector divided by speed, giving the direction of motion.
- Principal Normal
- The unit direction toward which the tangent is turning.
- Smooth Curve
- A parametrised curve with a continuous derivative that is nonzero on the interval considered.
- Arc-Length Parameter
- A parameter measuring distance along the curve, for which the derivative has unit magnitude.
- Curvature
- The nonnegative rate at which the unit tangent changes with arc length.
- Osculating Circle
- The local circle with radius 1/κ and centre one curvature radius along the principal normal.
- Areal Velocity
- The area swept by a radius vector per unit time, equal to one half of ||r×v||.
Vector Motion, Arc Length and Curvature FAQ
What is the difference between velocity and speed?
Velocity is a tangent vector carrying direction and rate, while speed is its nonnegative magnitude. A particle can move at constant speed and still accelerate because its velocity direction changes. Uniform circular motion is the standard example: speed is constant, tangential acceleration is zero, but inward normal acceleration has magnitude v² divided by the radius.
Why use the cross-product curvature formula?
It combines velocity and acceleration without differentiating a normalised tangent, often reducing algebra. The denominator requires nonzero speed, and the magnitude makes curvature nonnegative. The formula returns only the amount of bending. To locate the centre of curvature, compute the principal normal separately and move a distance 1/κ from the point along that direction.
How is an osculating circle checked?
Its radius is the reciprocal of curvature, its centre lies along the principal normal, and it shares the curve's tangent at the point. Check that T and N are unit and perpendicular before using r+(1/κ)N. A centre on the convex side signals a normal-direction error; a negative radius signals that signed curvature has been mixed with the course's nonnegative convention.
Exam move
Use a fixed derivative ladder: write r, then v=r′, speed ||v||, and a=r″ on separate lines. Mark the common domain and any parameter where speed vanishes before dividing by ||v||. For geometry, calculate T, N and κ with the formula that creates the least algebra, then sketch the concave side before placing an osculating centre. For dynamics, split acceleration into tangential and normal parts and check units.
Rehearse one problem in which acceleration is integrated from initial data and one central-force argument showing r×v is constant. Keep vector answers and scalar magnitudes visibly separate so a correct number never substitutes for a requested direction.
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