Motion — Question 3

Question 3

One path label does not always mean a straight line

Three tracked points move in a fixed laboratory:

• A bead stays on a straight rail. It may pause or reverse, but never leaves the rail.

• B stays on a known curved rail shaped as a semicircle. It may also pause or reverse.

• C moves freely over a flat table; it is not confined to a known curve.

In this chapter, a straight-line model means that the tracked point remains on one fixed straight line in the chosen reference. “One degree of freedom” here means that one independent spatial label suffices to locate the point under its constraint. This need not be the same as a straight-line path.

1. Which systems satisfy the stated straight-line model? Does A cease to be a straight-line motion when it reverses?

2. Explain why distance measured along B’s known rail can locate B using one label, even though its path is not a straight line in the laboratory.

3. Why does C generally need two independent location labels? Can a single east–west ruler reading locate it uniquely on the table?

4. A camera records only C’s shadow on an east–west line. The shadow moves on that line. Does this prove that C itself moves along a straight line? Give two distinct table locations that would have the same shadow in an ideal perpendicular projection.

Question 3 - Solution

One path label does not always mean a straight line

(a) Path geometry is separate from direction of travel.

$$\text{A: }(x,y)=(x(t),y_0),\qquad x(t)\text{ may increase or decrease.}$$

Only $\boxed{\text{A}}$ is guaranteed to stay on a fixed straight line. Pausing or reversing changes the time sequence of its locations, not the set of permitted locations. B’s constraint is curved, and C has no straight-line constraint.

(b) A known curved constraint can still leave one degree of freedom.

For a semicircle of radius $R$ centred at $(x_c,y_c)$, one possible parametrization is $$\text{B: }(x,y)=(x_c+R\cos\theta,\ y_c+R\sin\theta),\quad0\leq\theta\leq\pi.$$

Choose one end of B’s rail as the start. Distance measured along that rail identifies one location on this nonbranching semicircle. One independent label suffices because the rail supplies the missing geometric relation. Nevertheless, no single fixed straight line contains the whole path. A one-parameter constraint and a straight-line path are different claims.

(c) One projected label leaves a transverse ambiguity.

$$\text{C: }(x,y)=(x(t),y(t));\quad (2,1)\ne(2,3)\ \text{but both project to }x=2.$$

For C, an east–west location does not determine its north–south location. In general both are needed. The table constrains the point to a plane but does not constrain it to a particular curve within that plane.

(d) A projection can conceal additional motion.

For example, locations 2 m east and 1 m north of a corner, and 2 m east and 3 m north of it, have the same east–west shadow in the stipulated perpendicular projection. C could move between them while its shadow remained fixed, or combine north–south and east–west motion while the shadow moved along the ruler.

A one-dimensional record may be only one component of a richer motion. To justify a straight-line model, use a physical constraint or observations that test the omitted directions, not merely the shape of a projected trace.