AP Calculus BC glossary

Motion in the plane

Also called: Planar motion, Two-dimensional motion

Motion in the plane describes a particle whose position is a vector, x of t and y of t. Velocity is the vector x prime of t and y prime of t, speed is its magnitude, square root of x prime squared plus y prime squared, and total distance is the integral of speed over the time interval.

It is the planar version of straight-line motion. The position is ⟨x(t),y(t)⟩\langle x(t), y(t)\rangle, and you differentiate each component on its own: velocity is ⟨x′(t),y′(t)⟩\langle x'(t), y'(t)\rangle and acceleration is ⟨x′′(t),y′′(t)⟩\langle x''(t), y''(t)\rangle.

speed=(x′(t))2+(y′(t))2,distance=∫ab(x′(t))2+(y′(t))2 dt\text{speed} = \sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}, \qquad \text{distance} = \int_a^b \sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}\,dt

The mistake

Finding total distance by integrating each component separately and combining afterward. Distance is the integral of the speed, the single scalar (x′(t))2+(y′(t))2\sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}, formed before integrating. A particle is at rest only when x′(t)x'(t) and y′(t)y'(t) are both zero at the same instant.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)