Problem 34 worked answer

Least Action Lab

Designed for Grade 7 · Uses Middle Grade 6 math · About 8–16 minutes

Complete to collectTumbled JadeLevel 2

At a glance

History A: step actions $(4, 4, 4)$ ; total action $12$ .
History B: step actions $(1, 9, 4)$ ; total action $14$ .
History C: step actions $(5, 4, 5)$ ; total action $14$ .
History D: step actions $(5, 8, 5)$ ; total action $18$ .
Least action: History A.
Equal-action surprise: Histories B and C both have total action $14$ .
Motion selected by this model: straight motion with equal two-unit changes each second.

Key idea

Action belongs to a whole timed history, not only to the geometric line the puck traces. Every one-second change contributes a squared amount. A large change in one second therefore costs more action than spreading the same overall motion evenly.

Worked solution, part by part

1. Complete and compare the action table

For each ordered pair $(Δ 𝑥, Δ 𝑦)$ , substitute the two changes into

𝐴 s t e p = (Δ 𝑥) 2 + (Δ 𝑦) 2 .

History	0–1 s	1–2 s	2–3 s	Total action
A	$22 + 02 = 4$	$22 + 02 = 4$	$22 + 02 = 4$	$12$
B	$12 + 02 = 1$	$32 + 02 = 9$	$22 + 02 = 4$	$14$
C	$22 + 12 = 5$	$22 + 02 = 4$	$22 + (- 1) 2 = 5$	$14$
D	$22 + (- 1) 2 = 5$	$22 + 22 = 8$	$22 + (- 1) 2 = 5$	$18$

The complete comparison is

12 < 14 = 14 < 18 .

Therefore History A has the least total action among the four supplied histories.

2. Explain the equal-action histories

Histories B and C both have total action $14$ , but they arrive there differently.

History B stays on the horizontal line $𝑦 = 2$ . Its one-second horizontal changes are 1, 3, and 2, so its step actions are $1, 9, 4$ .
History C bends upward and then downward. Its step changes give actions $5, 4, 5$ .

The addends are different, but

1 + 9 + 4 = 5 + 4 + 5 = 14 .

This tie shows why action is not merely path length or straightness. The timing of the motion also matters.

3. Describe the least-action motion and the principle

History A stays on $𝑦 = 2$ , so its direction never bends. Its positions are spaced two horizontal units apart at every one-second mark. The puck therefore moves in a straight line at a steady rate.

A complete model sentence is: In this free-puck model, the principle of least action predicts the candidate history with the smallest sum of step actions.

Equivalent wording is acceptable if it says that complete possible histories are compared and the least-action one is selected in this model.

Check and completeness

Each history starts at $(0, 2)$ , ends at $(6, 2)$ , and uses three one-second changes. Adding the three change pairs in every row gives the same net change $(6, 0)$ , so the comparison holds the endpoints and elapsed time fixed.

All four supplied candidates have been evaluated. Since $12$ is smaller than both $14$ totals and $18$ , the claim that A is least among these four is complete rather than based on appearance.

Teaching note: why the simplified score is physical

For a free particle, the Lagrangian is its kinetic energy. Over equal time intervals, with the same mass for every candidate, the action is a common positive factor times the sum of the squared step lengths. Removing that common factor leaves the ranking unchanged and produces the worksheet's score.

The name principle of least action is traditional and is exact for this free-particle example. In general mechanics, the more precise statement is that the physical history makes the action stationary; it need not be the absolute smallest value.

Technical fit and rating

MJ 6.5 · C2 · W2 Standard challenge · Moderate workload

Algebraic structure and equations

How the rating works →

Worksheet arc · Introduction · 1 of 3 on Complete arc

Discover Action

Compare a small set of complete timed motions and notice that both route and timing affect action.