THERMO Spoken Here! ~ J. Pohl © | TOC NEXT ~ 26 |

At time, **t = 0**, all is "setup or ready" for the event. Geographic points "**A**" and "**B**" are located as shown.

At time, t = 0+, a cart commences to move with a speed of 2miles per hr from point "**A**" down the line **a - - a**. Eventually moving point "**A**" will pass through point **C** thereafter to continue in a straight line.

i) Calculate the least distance that occurs between the cart and point "**B**".

ii) When does this least distance occur?

**Solution:** The first step is to write a vector triangle that relates the changing position of the cart with the constant position of point "**B**".

(1) |

The above equation is written in an "implicit" (or inspecific) form. To proceed we make the equation explicit.
Of the three terms, the position of point "**B**," is easiest to write. It is known **P**_{B} is known.
The first, leftmost, term represents the position of the cart as a function of time, **P**(t)_{cart}. licit" do the easy parts first.

(2) |

Also the position of the Cart equals its initial position plus its constant velocity times time.

(3) |

We know the initial position of "A." We know part of the velocity of "A." We know its speed to be 2 miles per hour.

(4) |

Put this information into our equation.

(5) |

The direction of the velocity is the unit vector, e_{A}.
We can determine this direction by use of the triangle: 0AB.

(6) |

Entering the vectors and solve for AC:

(7) |

(9) |

(10) |

(11) |

Next we substitute the unit vector direction of movement of "**A**" into its eqation. Note we omit units in this equation.

(12) |

At time, **t = 0**, a cart and locations "**A**" and "**B**" are shown.

At time, t = 0+, the cart move with a speed of 2 miles per hr from point "**A**" down the line **a - - a**. The cart continues to move through point **C** on its straight line path.

i) Calculate the least distance that occurs between the cart and point "**B**".

ii) Determine when this least distance occur?

unwritten...