Vector functions in the plane

Differentiation component by component

• If the position vector is r(t) = x(t) i + y(t) j, its derivative is found by differentiating each component: v = dr/dt = ẋ i + ẏ j and a = dv/dt = ẍ i + ÿ j.
• The magnitude of velocity is |v| = √(ẋ² + ẏ²); the magnitude of acceleration is found the same way.
• Integration is also done component by component, and each component needs its own constant — together they form a constant vector.

Worked example

1. A particle moves so that r = (t³ − 3t) i + 2t² j. Then v = (3t² − 3) i + 4t j and a = 6t i + 4 j.
   At t = 2: v = 9 i + 8 j and a = 12 i + 4 j.
   |v| = √(81 + 64) = √145 ≈ 12.04 m/s.
2. Path elimination: from x = t³ − 3t and y = 2t² the trajectory can be studied by eliminating t; differentiating is usually far quicker than finding the cartesian path.