Worksheets · Calculus

Motion along a line worksheet

Velocity is how fast the position is changing and acceleration is how fast the velocity is, so the three sit one derivative apart. A particle at rest is one whose velocity is zero, whatever its acceleration is doing.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Motion Along a Line

Date Period

Answer each question about the particle.

  1. A particle moves along a line with s(t)=4t3−42t2+72t+9, t≥0. Find the velocity at t=2
  2. A particle moves along a line with s(t)=−2t3+18t2−48t−7, t≥0. Find the acceleration at t=4
  3. A particle moves along a line with s(t)=2t3−15t2+24t−6, t≥0. When is it at rest?
  4. A particle moves along a line with s(t)=4t3−30t2+48t−4, t≥0. Find the velocity at t=2
  5. A particle moves along a line with s(t)=2t3−33t2+180t+4, t≥0. Find the acceleration at t=4
  6. A particle moves along a line with s(t)=−2t3+21t2−72t+4, t≥0. When is it at rest?
  7. A particle moves along a line with s(t)=−2t3+21t2−60t+2, t≥0. Find the velocity at t=8
  8. A particle moves along a line with s(t)=2t3−30t2+96t−4, t≥0. Find the acceleration at t=6
  9. A particle moves along a line with s(t)=−4t3+48t2−84t, t≥0. When is it at rest?
  10. A particle moves along a line with s(t)=−2t3+30t2−96t−7, t≥0. Find the velocity at t=4
  11. A particle moves along a line with s(t)=4t3−54t2+216t−8, t≥0. Find the acceleration at t=8
  12. A particle moves along a line with s(t)=−2t3+30t2−126t+7, t≥0. When is it at rest?
  13. A particle moves along a line with s(t)=4t3−30t2+72t−3, t≥0. Find the velocity at t=1
  14. A particle moves along a line with s(t)=−4t3+18t2−24t+3, t≥0. Find the acceleration at t=1
  15. A particle moves along a line with s(t)=2t3−21t2+60t+9, t≥0. When is it at rest?
  16. A particle moves along a line with s(t)=4t3−66t2+288t−8, t≥0. Find the velocity at t=2
  17. A particle moves along a line with s(t)=−2t3+27t2−84t−2, t≥0. Find the acceleration at t=5
  18. A particle moves along a line with s(t)=−4t3+66t2−288t+7, t≥0. When is it at rest?
  19. A particle moves along a line with s(t)=−2t3+24t2−72t−6, t≥0. Find the velocity at t=2
  20. A particle moves along a line with s(t)=−2t3+30t2−96t−9, t≥0. Find the acceleration at t=7

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Motion Along a LineAnswer keyVersion 1

Date Period

Answer each question about the particle.

  1. A particle moves along a line with s(t)=4t3−42t2+72t+9, t≥0. Find the velocity at t=2−48
  2. A particle moves along a line with s(t)=−2t3+18t2−48t−7, t≥0. Find the acceleration at t=4−12
  3. A particle moves along a line with s(t)=2t3−15t2+24t−6, t≥0. When is it at rest?t=1, t=4
  4. A particle moves along a line with s(t)=4t3−30t2+48t−4, t≥0. Find the velocity at t=2−24
  5. A particle moves along a line with s(t)=2t3−33t2+180t+4, t≥0. Find the acceleration at t=4−18
  6. A particle moves along a line with s(t)=−2t3+21t2−72t+4, t≥0. When is it at rest?t=3, t=4
  7. A particle moves along a line with s(t)=−2t3+21t2−60t+2, t≥0. Find the velocity at t=8−108
  8. A particle moves along a line with s(t)=2t3−30t2+96t−4, t≥0. Find the acceleration at t=612
  9. A particle moves along a line with s(t)=−4t3+48t2−84t, t≥0. When is it at rest?t=1, t=7
  10. A particle moves along a line with s(t)=−2t3+30t2−96t−7, t≥0. Find the velocity at t=448
  11. A particle moves along a line with s(t)=4t3−54t2+216t−8, t≥0. Find the acceleration at t=884
  12. A particle moves along a line with s(t)=−2t3+30t2−126t+7, t≥0. When is it at rest?t=3, t=7
  13. A particle moves along a line with s(t)=4t3−30t2+72t−3, t≥0. Find the velocity at t=124
  14. A particle moves along a line with s(t)=−4t3+18t2−24t+3, t≥0. Find the acceleration at t=112
  15. A particle moves along a line with s(t)=2t3−21t2+60t+9, t≥0. When is it at rest?t=2, t=5
  16. A particle moves along a line with s(t)=4t3−66t2+288t−8, t≥0. Find the velocity at t=272
  17. A particle moves along a line with s(t)=−2t3+27t2−84t−2, t≥0. Find the acceleration at t=5−6
  18. A particle moves along a line with s(t)=−4t3+66t2−288t+7, t≥0. When is it at rest?t=3, t=8
  19. A particle moves along a line with s(t)=−2t3+24t2−72t−6, t≥0. Find the velocity at t=20
  20. A particle moves along a line with s(t)=−2t3+30t2−96t−9, t≥0. Find the acceleration at t=7−24

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

A particle moves with s(t)=t3−6t2+9t. When is it at rest?

  1. The velocity is 3t squared minus 12t plus 9.
  2. That factors to 3 times t minus 1 times t minus 3.
  3. So it is at rest at t equals 1 and t equals 3.

The answer is t=1, t=3.

Where students go wrong

Setting the acceleration to zero to find where the particle stops. Zero acceleration means the velocity is not changing, which is the opposite: a particle with zero acceleration is usually moving at a steady speed.

Also called particle motion, position velocity acceleration, rectilinear motion or when is the particle at rest.

What you can put on this worksheet

Velocity at a moment
A particle moves along a line with s(t)=2t2−20t−8, t≥0. Find the velocity at t=4 → −4
Acceleration at a moment
A particle moves along a line with s(t)=2t2−20t−8, t≥0. Find the acceleration at t=4 → 4
When the particle is at rest
A particle moves along a line with s(t)=2t2−20t−8, t≥0. When is it at rest? → t=5

Questions about these worksheets

Yes. Every motion along a line sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the motion along a line questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: velocity at a moment, acceleration at a moment and when the particle is at rest. Tick as many as you want and set how many of each, or let it spread them evenly.

Setting the acceleration to zero to find where the particle stops. Zero acceleration means the velocity is not changing, which is the opposite: a particle with zero acceleration is usually moving at a steady speed.