Discrete and continuous modelling

Hi there,

I am currently stuck on this question.

A woman is running along the line x = 50 (assuming constant speed)
Her dog returns a ball to her but runs in a curved line, as the dog runs towards wherever she is at any instant. To simplify the problem, we assume that the dog starts his journey to her from the origin of suitable rectangular axes, whilst she sets off on her run from (50,0) at the instant the dog starts his run with the ball from the origin.

I am required to formulate a discrete model and a continuous model, and an equation involving the dogs position, the gradient of its location in the x/y plane and the position of the woman, this is all in order to find the position and displacement of the dog at any given time.

Thank you, I will update my post if I make any advance.

At any time t, the woman's position is (50, vt) where v is her constant speed. Suppose the dog's position at time t is (x(t), y(t)). Then the slope of the line from the dog's position to the woman is (vt- y)/(50- x). That is equal to the derivative of y with respect to x.

I believe this has something to do with tractrix.

Thank you HallsofIvy, great help and fast reply!!

Emakarov, I don't think tractrix apply here seeing as the woman is travelling in a straight line and the dog heads to wherever she is at any given point. Thank you for the idea though.

I had a brief look at this again yesterday and came to the conclusion that if I were to take the dogs position at any given point, the horizontal displacement (x) would be equal to (t)(U)cos(theta).

I'm going to try and incorporate this info into a discrete and a continuous model and I will update tomorrow. Goodnight.

Quote:

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Originally Posted by CrazyAbe

Emakarov, I don't think tractrix apply here seeing as the woman is travelling in a straight line and the dog heads to wherever she is at any given point.

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This is precisely the property of a tractrix. The curve in your case is a curve of pursuit; however, you are right that it is not necessarily a tractrix. Among curves of pursuit, a tractrix has the following special properties:

(1) the leader moves at a constant velocity,
(2) this velocity forms a right angle to the initial line between the leader and the pursuer, and
(3) the distance between the leader and the pursuer is constant.

From the problem statement, property (3) does not necessarily hold in your case.