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Worksheets›Calculus›Applications of derivatives›Position, velocity and acceleration

Position, Velocity and Acceleration Worksheets

Free printable worksheets with answer keys. Pick a level, preview the questions, then open the sheet to swap or delete any single question.

CalculusAnswer key includedPDF · Print
Open in generatorStart with an easy sheet

Easy: Velocity at a time

  1. 1.s(t) = 3t3 − t2 + 5tFind the velocity at t = 5.
  2. 2.s(t) = −2t3 − 5t2 − 9tFind the velocity at t = 5.
  3. 3.s(t) = t3 + 3t2 − 10tFind the velocity at t = 3.
  4. 4.s(t) = t3 + 4t2 − tFind the velocity at t = 5.
  5. 5.s(t) = −2t3 − 7t2 − 9tFind the velocity at t = 3.
  6. 6.s(t) = 3t3 − 9tFind the velocity at t = 2.
Preview shows 6 of 12 questions. Every sheet is new; open it to change any question.

Worksheet variations

Velocity is the derivative of position, v(t) = s′(t), and acceleration is the derivative of velocity, a(t) = v′(t) = s″(t). The object is at rest when v(t) = 0, moving forward when v(t) > 0 and backward when v(t) < 0.

Easys(t) = t3 − 6t2 − 10tFind the velocity at t = 4.Velocity at a timeMediums(t) = −3t3 − 4t2 + 4tFind the acceleration at t = 5.Acceleration at a timeHards(t) = t3 − 15t2 + 72tWhen is the object at rest?When the object is at rest

How to solve it: a worked example

s(t) is the position of an object in meters after t seconds.

  1. Starts(t) = −3t3 − 2t2Find the velocity at t = 3.
  2. Velocity is the derivativev(t) = −9t2 − 4t
  3. Substitute the timev(3) = −93
  4. Answer−93 m/s

Common mistakes to watch for

  • Substituting into s(t) when the question asks for velocity.
  • Differentiating only once for acceleration.
  • Including a negative time as a time at rest.

Tip: tick “Step-by-step answer key” in the generator to print each step for marking.

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FAQ

How are position, velocity and acceleration related?

Velocity is the derivative of position, and acceleration is the derivative of velocity. Each tells how fast the one before it is changing.

What does it mean when velocity is 0?

The object is momentarily at rest. If the velocity changes sign there, the object turns around.

Is every worksheet different?

Yes. Numbers are generated each time, and the answer key is calculated for that sheet. Swap any single question and its answer updates too.

Can I mix position, velocity and acceleration with other skills?

Yes. In the generator, add more skills; each gets its own section on the sheet with its own instructions.

Is it free to print?

Yes. Generating and printing from the browser is free with no sign-up. PDF downloads need a free account.