Classical Mechanics (Kibble & Berkshire) · Problem 4.26

Linear oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · projectile-linear-resistance-general

Problem

Show that, when a particle is projected from the origin in a medium that exerts linear resistance, its position vector at time tt has the general form r=−α(t)k+β(t)u,\boldsymbol{r} = -\alpha(t)\boldsymbol{k} + \beta(t)\boldsymbol{u}, where k\boldsymbol{k} is the vertically upwards unit vector and u\boldsymbol{u} is the velocity of projection. Deduce the following results: ((i)) A number of particles are projected simultaneously from the same point, with the same speed, but in different directions. Show that, at each later time, the particles all lie on the surface of a sphere. ((ii)) A number of particles are projected simultaneously from the same point, in the same direction, but with different speeds. Show that, at each later time, the particles all lie on a straight line. ((iii)) Three particles are projected simultaneously in a completely general manner. Show that the plane containing the three particles remains parallel to some fixed plane.
Intuitive explanation

Linear resistance fundamentally decouples the vertical and horizontal motion: gravity acts uniformly downward while drag proportionally opposes velocity in any direction. The key insight is that the equation of motion mr¨=−mgk−mγr˙m\ddot{\boldsymbol{r}} = -mg\boldsymbol{k} - m\gamma\dot{\boldsymbol{r}} separates into independent scalar functions of time multiplying the constant vectors k\boldsymbol{k} and u\boldsymbol{u}—this structure persists through integration and reveals hidden geometric symmetries in how particle clouds evolve.

Intuitive explanation

Linear resistance fundamentally decouples the vertical and horizontal motion: gravity acts uniformly downward while drag proportionally opposes velocity in any direction. The key insight is that the equation of motion mr¨=−mgk−mγr˙m\ddot{\boldsymbol{r}} = -mg\boldsymbol{k} - m\gamma\dot{\boldsymbol{r}} separates into independent scalar functions of time multiplying the constant vectors k\boldsymbol{k} and u\boldsymbol{u}—this structure persists through integration and reveals hidden geometric symmetries in how particle clouds evolve.

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