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CalculusBy Educator Today

C40 · Unit C9 Parametric, polar and vector calculus

Vector integration and motion in the plane

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 9.5–9.6 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Integrate vector velocity componentwise and use an initial position to determine both constants. Acceleration is the componentwise derivative of velocity. Speed increases when v times a>0 and decreases when v times a<0, provided speed is nonzero; this follows by differentiating |v| squared equals v times v. A particle can have constant speed while its direction and velocity change.

Worked example

With v(t) equals 〈2t,3〉 and r(0) equals 〈1, minus 2〉, position is r(t) equals 〈t squared +1,3t minus 2〉. At t equals 2 it is 〈5,4〉. Acceleration is 〈2,0〉; v times a equals 4t, so speed increases for t>0.

Explore the mathematics

Vector integration and motion in the plane

Integrate two velocity components to reconstruct a path.

Setting 3 of 5

Scroll sideways to see the full diagram.

How do two velocity components determine a path?xy123456789Velocity v(t)=(2t,1); initial position (0,0)

Predict before continuing

How do two velocity components determine a path?

Read the complete explanation
  1. How do two velocity components determine a path?
  2. Integrate each component and apply both initial coordinates.
  3. Position is r(t)=(t²,t), with both integration constants fixed at 0.
  4. At t=1.5, position=(2.25,1.5), velocity=(3,1) and speed=√(3²+1)=3.1623.
See the idea: Vector integration and motion in the plane
See the idea: Vector integration and motion in the planeFor t from 0 to 2, r(t)=(t²+1,3t−2) traces this path from (1,−2) to (5,4). Position is not velocity: v=(2t,3) and acceleration is (2,0).xy1-22-0.2531.543.2555t=0t=2

For t from 0 to 2, r(t)=(t²+1,3t−2) traces this path from (1,−2) to (5,4). Position is not velocity: v=(2t,3) and acceleration is (2,0).

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Integrating speed cannot recover direction or position. Each position component needs its own initial value.

Investigate and explain

Plot a vector trajectory and its velocity arrows. Distinguish where the particle is from where it is heading.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.