# Rate problems

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### Uniform circular motion

A particle is said to be undergoing uniform circular motion if its radius vector in appropriate coordinates has the form , where(1)(2)Geometrically, uniform circular motions means that moves in a circle in the -plane with some radius at constant speed. The quantity is called the angular velocity of . The speed of is(3)and the acceleration of P has constant magnitude(4)and is directed toward the center of the circle traced by . This is called centripetal acceleration.Ignoring the ellipticity of their orbits, planet show nearly uniform circular motion about the Sun. (Although due to orbital inclinations, the orbital planes of the different planets are not necessarily coplanar.)

### Two trains puzzle

Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h. Upon reaching the other train, the fly turns around and continues towards the first train. How many kilometers does the fly travel before getting squashed in the collision of the two trains?Now, the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time. The position of the fly at time is plotted above.However, a brute force method instead solves for the position of the fly along each traversal between the trains. For example, the fly reaches the second train when(1)or h, at which point..

### Trawler problem

A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..

### Tractrix

The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog's master, the curve has the descriptive name "hundkurve" (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve.In..

A problem in the calculus of variations. Let a vessel traveling at constant speed navigate on a body of water having surface velocity(1)(2)The navigation problem asks for the course which travels between two points in minimal time.

### Mice problem

In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...

### Brocard points

The first Brocard point is the interior point (also denoted or ) of a triangle with points labeled in counterclockwise order for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(1)(Kimberling 1998, p. 47).Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.The second Brocard point is the interior point (also denoted or ) for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(2)(Kimberling 1998, p. 47).Moreover, the two angles are equal, and this angle is called the Brocard angle,(3)(4)The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they..