Worksheets · Calculus

Motion in the plane worksheet

Each coordinate is differentiated on its own: the velocity vector is ⟨x′(t), y′(t)⟩ and the acceleration vector is ⟨x″(t), y″(t)⟩. Speed is the length of the velocity vector.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Motion in the Plane

Date Period

Answer each question about the particle.

  1. A particle moves with x(t)=t2 and y(t)=3t3. Find its speed at t=1
  2. A particle moves with x(t)=2t2+5t and y(t)=−2t3. Find its speed at t=1
  3. A particle moves with x(t)=−t2 and y(t)=−2t3. Find its velocity vector at t=3
  4. A particle moves with x(t)=−t2−t and y(t)=3t3. Find its velocity vector at t=1
  5. A particle moves with x(t)=2t2+3t and y(t)=−3t3. Find its acceleration vector at t=2
  6. A particle moves with x(t)=t2+3t and y(t)=−t3. Find its velocity vector at t=2
  7. A particle moves with x(t)=−3t2+t and y(t)=−t3. Find its speed at t=1
  8. A particle moves with x(t)=t2+5t and y(t)=2t3. Find its acceleration vector at t=1
  9. A particle moves with x(t)=3t2+2t and y(t)=3t3. Find its velocity vector at t=3
  10. A particle moves with x(t)=−3t2−2t and y(t)=2t3. Find its acceleration vector at t=3
  11. A particle moves with x(t)=−3t2+5t and y(t)=−3t3. Find its speed at t=1
  12. A particle moves with x(t)=−2t2−2t and y(t)=2t3. Find its velocity vector at t=2
  13. A particle moves with x(t)=3t2−2t and y(t)=−t3. Find its velocity vector at t=3
  14. A particle moves with x(t)=−2t2+t and y(t)=−t3. Find its velocity vector at t=1
  15. A particle moves with x(t)=−3t2+2t and y(t)=−2t3. Find its velocity vector at t=2
  16. A particle moves with x(t)=t2+5t and y(t)=3t3. Find its velocity vector at t=1
  17. A particle moves with x(t)=−3t2−2t and y(t)=3t3. Find its velocity vector at t=1
  18. A particle moves with x(t)=3t2+3t and y(t)=−t3. Find its speed at t=1
  19. A particle moves with x(t)=t2+4t and y(t)=−3t3. Find its speed at t=3
  20. A particle moves with x(t)=t2−t and y(t)=−t3. Find its acceleration vector at t=1

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

Sigma Prep · sigmaprep.io/worksheets

Name

Motion in the PlaneAnswer keyVersion 1

Date Period

Answer each question about the particle.

  1. A particle moves with x(t)=t2 and y(t)=3t3. Find its speed at t=185
  2. A particle moves with x(t)=2t2+5t and y(t)=−2t3. Find its speed at t=1313
  3. A particle moves with x(t)=−t2 and y(t)=−2t3. Find its velocity vector at t=3⟨−6,−54⟩
  4. A particle moves with x(t)=−t2−t and y(t)=3t3. Find its velocity vector at t=1⟨−3,9⟩
  5. A particle moves with x(t)=2t2+3t and y(t)=−3t3. Find its acceleration vector at t=2⟨4,−36⟩
  6. A particle moves with x(t)=t2+3t and y(t)=−t3. Find its velocity vector at t=2⟨7,−12⟩
  7. A particle moves with x(t)=−3t2+t and y(t)=−t3. Find its speed at t=134
  8. A particle moves with x(t)=t2+5t and y(t)=2t3. Find its acceleration vector at t=1⟨2,12⟩
  9. A particle moves with x(t)=3t2+2t and y(t)=3t3. Find its velocity vector at t=3⟨20,81⟩
  10. A particle moves with x(t)=−3t2−2t and y(t)=2t3. Find its acceleration vector at t=3⟨−6,36⟩
  11. A particle moves with x(t)=−3t2+5t and y(t)=−3t3. Find its speed at t=182
  12. A particle moves with x(t)=−2t2−2t and y(t)=2t3. Find its velocity vector at t=2⟨−10,24⟩
  13. A particle moves with x(t)=3t2−2t and y(t)=−t3. Find its velocity vector at t=3⟨16,−27⟩
  14. A particle moves with x(t)=−2t2+t and y(t)=−t3. Find its velocity vector at t=1⟨−3,−3⟩
  15. A particle moves with x(t)=−3t2+2t and y(t)=−2t3. Find its velocity vector at t=2⟨−10,−24⟩
  16. A particle moves with x(t)=t2+5t and y(t)=3t3. Find its velocity vector at t=1⟨7,9⟩
  17. A particle moves with x(t)=−3t2−2t and y(t)=3t3. Find its velocity vector at t=1⟨−8,9⟩
  18. A particle moves with x(t)=3t2+3t and y(t)=−t3. Find its speed at t=1310
  19. A particle moves with x(t)=t2+4t and y(t)=−3t3. Find its speed at t=36661
  20. A particle moves with x(t)=t2−t and y(t)=−t3. Find its acceleration vector at t=1⟨2,−6⟩

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

How to do these

x(t)=t2, y(t)=t3. Find the speed at t=1

  1. The velocity at t = 1 is ⟨2, 3⟩.
  2. Speed is the square root of 4 + 9.

The answer is 13.

Where students go wrong

Adding the two velocity components to get the speed. Speed is the length of the vector, so the components are squared, added and square-rooted.

Also called vector-valued functions, particle motion in the plane or parametric motion.

Questions about these worksheets

Yes. Every motion in the plane 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 in the plane 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.

Adding the two velocity components to get the speed. Speed is the length of the vector, so the components are squared, added and square-rooted.