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Problem 41

Action Between Frames

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Action Between Frames

Complete to collectAlexandrite CrystalLevel 2 · 4 Glints

The camera should not change the motion

A puck travels from at to at . A camera divides the same one-second history into equal time intervals.

If the camera records more frames, should the action of the motion itself change?

One motion, several measurements

The history stays fixed

The puck still travels from 0 to 1 in one second.

The frame count changes

divides that same second more finely.

Each duration changes

For equal slices, .

If a proposed action changes only because the camera took more frames, it is measuring the recording scheme—not only the motion.

Two position-versus-time graphs sampled at four equal time intervals. The steady history x equals t has equal position changes. The accelerating history x equals t squared has increasingly large position changes.
Equal time slices reveal how the speed changes along a history.

Page 1 · Test the tempting score

Complete the tables here. Show calculus work on separate paper.

  1. First test the tempting score on the steady history . Complete the table. The physical motion is unchanged, so explain why the tempting score cannot be its action.

    Each
    111
    2
    4

Page 2 · Repair the score and take the limit

Complete the table here. Show calculus work on separate paper.

Action between frames

Each interval contributes speed squared × duration. The factor accounts for how long that speed lasts.

  1. Now use for . Show in one line that the corrected score equals 1 for every . Then complete the accelerating-history table for .

    Interval speeds
    11
    2
    4

  2. In the limit of infinitely fine time slices, the sums become

    Compute both histories' continuous actions. Compare them with the frame sums. Explain what changes under finer measurement and what stays attached to the history.

Normalization note: A fixed positive factor does not change comparisons; hid it in Lab 1.

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About this problem

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Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 12

Uses HS CALC math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 12–24 minutes

Moderate amount of written work · Short Explanation response.

How to support: The key question is whether recording the same motion with more frames should change its action. Ask what each time slice contributes, without supplying the corrected totals.

Worksheet arc · Guided Practice · 2 of 3 on Complete arc

Discover Action

Repair the discrete score so it is independent of camera resolution, then recognize its continuous integral.

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Technical reference

Rating, skills & curriculum

Designed for Grade 12 · Uses HS CALC math · About 12–24 minutes

Grade 12 · Calculus and continuous change · Integrals and the Fundamental Theorem · Limits and continuity

MJ HS:CALC.8 · C2 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:CALC.8

High school

HS CALC · point 8

Likely range
MJ HS:CALC.6–HS:CALC.8
Confidence
72%

Challenge

C2 · Standard

The solver chooses and coordinates familiar methods.

  • Method selection1
  • Reasoning depth2
  • Novelty2
  • Constraint management1
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
12–24 minutes · typically 17
Response
Short Explanation
Known-method steps
17
  • Mechanical execution2
  • Written output2
  • Representation production1
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Interpret definite integrals as accumulation and signed area

    The learner can interpret definite integrals as accumulation and signed area. This capability is distinct from mere exposure to or isolated use of limits; area; functions.

    • Developing mastery
    • 70% focus
    • MJ HS:CALC.5 milestone

  • Understand a limit as local or end behavior of a function

    The learner can understand a limit as local or end behavior of a function. This capability is distinct from mere exposure to or isolated use of functions; approximation.

    • Independent mastery
    • 30% focus
    • MJ HS:CALC.2 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Compute derivatives using standard rules

    The learner can compute derivatives using standard rules. This capability is distinct from mere exposure to or isolated use of algebra; derivative definition.

    • Fluent readiness
    • Required
    • MJ HS:CALC.6 milestone

  • Find basic antiderivatives and indefinite integrals

    The learner can find basic antiderivatives and indefinite integrals. This capability is distinct from mere exposure to or isolated use of derivative rules.

    • Fluent readiness
    • Required
    • MJ HS:CALC.8 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Practice locations1 location · Representative: Construct viable arguments and critique the reasoning of others.