Example 1

Curvilinear motion is an object moving in a plane along a specific path. To find the *velocity* of an object whose coordinates are given in parametric form, find its x-component of velocity
by determining dx/dt and its y-component of velocity by determining dy/dt.

Once these are evaluated at a specific time, you can then find the resultant velocity from

The direction, , in which the object is moving is found from

Similarly for *acceleration:* x-component of acceleration

Remember: | If s(t) = f(t) is a position or distance functions, |

then v(t) = s'(t) = is the velocity function, | |

and a(t) = s''(t) = is the acceleration function. |

Sometimes the x and y coordinates of a point on a curve are given as functions of t. This is parametric form and t is the parameter.

**Try it!**

Given

Find the magnitude and direction of the velocity when t = 1.

magnitude of the velocity:

direction of motion:

is in the 1st quadrant because and are both positive. |

Example 2

Find the magnitude and direction of (a) velocity vector and (b) acceleration vector at time t = 1.

magnitude of the velocity:

direction of motion:

is in the 4th quadrant because is positive and is negative. | |

magnitude of acceleration:

direction of acceleration:

is in the 1th quadrant because and are both positive. |