Motion — Question 10

Question 10

Two gates can reveal direction and size without a speed calculation

Two ideal narrow light gates lie on a fixed straight rail, A to the left of B, separated by $D=0.80\ \mathrm{m}$. A rigid opaque bar of unknown length $L$ stays parallel to the rail. In each run it starts fully outside the gate pair, passes completely across both gates once, and ends fully outside on the other side. It never reverses, rotates, or changes length during a run. Its pace may vary.

A gate turns ON when the leading end first blocks it and OFF when the trailing end clears it. Response delay and beam width are negligible. Recorded events are: Run X records A ON at $1.0\ \mathrm{s}$, B ON at $1.4\ \mathrm{s}$, A OFF at $1.8\ \mathrm{s}$ and B OFF at $2.2\ \mathrm{s}$. Run Y records A ON at $1.0\ \mathrm{s}$, A OFF at $1.2\ \mathrm{s}$, B ON at $1.6\ \mathrm{s}$ and B OFF at $1.8\ \mathrm{s}$.

1. Infer the direction of passage in each run. Would a single gate’s ON–OFF sequence alone determine that direction?

2. For each run, determine whether $L<D$, $L=D$, or $L>D$. Justify using the two middle events and the bar’s geometry, not a speed formula.

3. Could the same unchanged bar produce both records merely by changing its forward pace? Explain. What middle-event pattern would indicate $L=D$ in this ideal model?

4. Name two model failures that would require revisiting the inference in a real apparatus. Explain why these records do not by themselves establish a steady pace.

Question 10 - Solution

Two gates can reveal direction and size without a speed calculation

(a) Determine the direction from the first blocked gate.

Take $x_A=0$ and $x_B=D=0.80\ \mathrm{m}$. In both records A turns ON before B. For a complete nonreversing passage this gives $$x_{\mathrm{front}}:0\longrightarrow D,\qquad\boxed{\text{motion from left to right}}.$$ One gate’s ON–OFF pair alone would allow either approach direction.

(b) Compare the front and rear at one shared event.

When the front first reaches B, $x_{\mathrm{front}}=D$ and $x_{\mathrm{rear}}=D-L$. Therefore $$\begin{aligned} X:\ t_{B\mathrm{ON}}<t_{A\mathrm{OFF}}&\Rightarrow D-L<0 &&\Rightarrow\boxed{L>0.80\ \mathrm{m}},\\ Y:\ t_{A\mathrm{OFF}}<t_{B\mathrm{ON}}&\Rightarrow D-L>0 &&\Rightarrow\boxed{L<0.80\ \mathrm{m}}. \end{aligned}$$ X has an interval with both gates blocked; Y has an interval after A clears but before B becomes blocked. These conclusions use geometric order, not a constant-speed assumption.

(c) A pace change cannot reverse spatial event order.

The relevant front coordinates are $D$ at B ON and $L$ at A OFF. A monotone trajectory encounters them in their spatial order: $$\operatorname{sgn}(t_{A\mathrm{OFF}}-t_{B\mathrm{ON}}) =\operatorname{sgn}(L-D).$$ Thus changing the pace cannot make one unchanged rigid bar satisfy both $L>D$ and $L<D$. At the boundary, $$\boxed{L=D\ \Longrightarrow\ t_{A\mathrm{OFF}}=t_{B\mathrm{ON}}}.$$ The middle events coincide for ideal narrow gates.

(d) Identify which experimental assumptions may fail.

Unequal detector delays or finite beam widths change recorded event times. Rotation, deformation or reversal changes the relation between front and rear events. Any two are sufficient examples of model failure. Moreover, finitely many threshold events constrain only selected positions and times; variable forward motion between them can reproduce the same record. They do not establish steady pace.