Least Distances 2

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.

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(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)

Least Distances 2

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?