The hodograph is very useful for solving complicated problems in dynamics. By simple geometrical arguments students can directly obtain the answer to problems that would otherwise be complicated exercises in algebra. Although beyond the level of undergraduates, we also use the hodograph to calculate by variational geometrical techniques, the well-known brachistochrone curve, thus illustrating this approach.

William Rowan
, “
The Hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction
Proc. R. Ir. Acad.
James Clerk Maxwell, Matter and Motion (Dover, New York, 1991).
David and Judith Goodstein, Feynman’s Lost Lecture (W. W. Norton, New York, 1996).
If sin φ=sc, so that the two graphs just touch each other [see Fig. 5(b)], the slope of the ground at the intersection point is equal to the inverse of the slope of f1, because [dyground/dx]0=[df2/dy]0−1=[df1/dy]0−1=[(d/dy)sc2a1+y/a]0−1=(1+y0/a)/sc. If we call φ1 and φ2 the angles formed by the initial and final velocities, respectively, with the horizon we have that φ12c, and cos φ1=1+y0/acos φ2, where the last relation comes from the geometry of Fig. 4. With a little algebra we obtain tan φ2=(1+y0/a)/sccot φc. Therefore if φc is the oblique angle that satisfies sin φc=sc, the slope of the projectile trajectory is smaller than the slope of the ground at the landing point, which is impossible (because this would mean that the projectile arrives at this point from inside the ground). Thus, only the corresponding obtuse-angle solution can lead the projectile to this point. The same reasoning holds for values of φ such that sin φ is slightly less than sc. Both intersection points then can be reached by the obtuse-angle solution, and the nearest point can be reached by the oblique-angle solution only if dyground/dx⩽tan φ2.
The stationary point we calculated here corresponds to a minimum, because an infinitely deep well would lead to infinite time travel.
Because all expressions are written as functions of u, we try to optimize the inverse function of u(θ); namely, θ(u).
The end points of θ are θin=−π/2, and θfin=π/2, because these values lead to zero velocities when substituted in u=(1/λ)cos θ, the hodographic curve.
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