Page:Elementary Text-book of Physics (Anthony, 1897).djvu/69

 any contral force, will always lie in one plane—that containing its original direction of motion and that of the force.

Suppose that the moving particle, which starts fron the point $$A$$ (Fig. 21), and which in the time $$t$$ moves over the distance $$AB$$, $$D$$ is acted on at the point $$B$$ by an impulse directed toward the centre $$O$$, such that the particle is displaced toward $$O$$ in the next equal time-interval $$t$$ by the distance $$BE$$. If the particle were not acted on by the impulse at $$B$$ it would continue to move in the line $$AB$$, and at the end of the second time-interval $$t$$ it would reach the point $$D$$, the line $$BD$$ being equal to the line $$AB$$. Join $$OD$$, draw $$DC$$ parallel to $$OB$$, and $$EC$$ parallel to $$BD$$; connect $$BC$$. Now the line $$BC$$ is the resultant of the displacement $$BD$$, which the particle would have in consequence of its original motion, and the displacement $$BE$$, given to it by the impulse at $$B$$. It is therefore the path traversed by the particle in the second time-interval. But the triangles $$OCB$$ and $$ODB$$, being on the same base and between the same parallels, are equal; and the triangles $$OBD$$ and $$OAB$$, being on equal bases and having the same vertex $$O$$, are equal. Therefore the triangle $$OCB$$ and the triangle $$OAB$$, described in equal time-intervals, are equal. If now the intervals into which the whole time is divided become infinitely small, in the limit the broken line ABC approaches indefinitely near to a curve, and the areas swept out in equal times by the radius vector drawn to the curve are equal.

If a line be drawn tangent to the path at any point, and a perpendicular, $$p$$, drawn to it from the centre, the area swept out by the radius vector as the particle describes a small distance $$s = vt$$, where $$v$$ is its velocity and $$t$$ the time in which $$s$$ is described, is given by $$\tfrac{1}{2} vpt$$, and since the areas described in equal times are equal, the product $$vp$$ is constant for all parts of the path.

49. Central Force Proportional to the Radius Vector.— If the central force which acts on the particle be proportional to the distance of the particle from the centre, the path which it describes