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Understanding Motion worksheets

Analyze polynomial position for velocity, direction changes, direction intervals, and displacement.

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Differentiate position, analyze velocity signs on the stated domain, and answer each selected property.

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Each worked key keeps the defining constraint, derivative reasoning, exact result, and contextual units together.

Learning objective

What this activity builds

Use velocity and acceleration on a closed time domain while distinguishing signed velocity, speed, direction, rest, and displacement.

Worked example

Does every zero of velocity change direction?

No.

Direction changes only when velocity changes sign; an even-multiplicity zero can be a rest instant without a change.

Teaching notes

Make the practice count

  • Test velocity signs between roots.
  • For maximum velocity, check endpoints and where acceleration is zero.
  • Compute displacement as final position minus initial position.

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Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.1. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 12. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=4-10·t+t2012Set-up and work:2. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 8. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=-6-6·t+6·t2-2·t308Set-up and work:math-sheet.comPage 1 of 6
Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.3. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 12. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=7-48·t+18·t2-2·t3012Set-up and work:4. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 8. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=-3-6·t+6·t2-2·t308Set-up and work:math-sheet.comPage 2 of 6
Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.5. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 10. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=7+108·t-27·t2+2·t3010Set-up and work:6. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 10. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=-4·t+t2010Set-up and work:math-sheet.comPage 3 of 6
Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.7. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 10. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=-6+4·t-t2010Set-up and work:8. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 12. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=-20·t+2·t2012Set-up and work:math-sheet.comPage 4 of 6
Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.9. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 12. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=90·t-24·t2+2·t3012Set-up and work:10. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 6. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=4-2·t+t206Set-up and work:math-sheet.comPage 5 of 6
Math SheetNameScoreTeacherDateUnderstanding MotionAnalyze position, velocity, and acceleration on the stated closed time interval.11. degree 3A particle has position s(t) in meters at time t seconds on 0 <= t <= 6. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=4-6·t+6·t2-2·t306Set-up and work:12. degree 2A particle has position s(t) in meters at time t seconds on 0 <= t <= 10. Find: maximum velocity; when direction changes;when it moves left and right; displacement over the interval.s(t)=7-4·t+t2010Set-up and work:math-sheet.comPage 6 of 6