s(t) = -t^3 + 3t^2 + 2, 0 ≤ t ≤ 3

v(t) = -3t^2 + 6t; a parabola through (0, 0) and (2, 0), vertex (1, 3)

Precalculus motion along a line worksheet

In Precalculus velocity comes from a limit: the average velocity over a shrinking time interval. Students build the difference quotient for a position function, simplify it, and let h go to 0, which is the idea behind the derivative.

Also taught in Calculus, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Motion Along a Line Date Period Answer each question about the particle. A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 72 t + 9$ , $t \geq 0$ . Find the velocity at $t = 2$ A particle moves along a line with $s (t) = - 2 t^{3} + 18 t^{2} - 48 t - 7$ , $t \geq 0$ . Find the acceleration at $t = 4$ A particle moves along a line with $s (t) = 2 t^{3} - 15 t^{2} + 24 t - 6$ , $t \geq 0$ . When is it at rest? A particle's position is $s (t) = t^{3} - 6 t^{2} + 9 t$ . At $t = 4$ , is it speeding up or slowing down? The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? A particle moves along a line with $s (t) = - 4 t^{3} + 30 t^{2} - 72 t + 7$ , $t \geq 0$ . Find the velocity at $t = 2$ A particle moves along a line with $s (t) = - 2 t^{3} + 42 t^{2} - 288 t + 3$ , $t \geq 0$ . Find the acceleration at $t = 9$ A particle moves along a line with $s (t) = - 4 t^{3} + 48 t^{2} - 84 t$ , $t \geq 0$ . When is it at rest? A particle's position is $s (t) = t^{3} - 6 t^{2}$ . At $t = 5$ , is it speeding up or slowing down? The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? A particle moves along a line with $s (t) = - 4 t^{3} + 54 t^{2} - 168 t$ , $t \geq 0$ . Find the velocity at $t = 4$ A particle moves along a line with $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 9$ , $t \geq 0$ . Find the acceleration at $t = 3$ A particle moves along a line with $s (t) = 4 t^{3} - 36 t^{2} + 60 t + 1$ , $t \geq 0$ . When is it at rest? A particle's position is $s (t) = t^{3} - 9 t^{2} + 24 t$ . At $t = 0$ , is it speeding up or slowing down? The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? A particle moves along a line with $s (t) = 2 t^{3} - 27 t^{2} + 108 t$ , $t \geq 0$ . Find the velocity at $t = 9$ A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 144 t + 8$ , $t \geq 0$ . Find the acceleration at $t = 0$ A particle moves along a line with $s (t) = - 2 t^{3} + 33 t^{2} - 168 t - 2$ , $t \geq 0$ . When is it at rest? A particle's position is $s (t) = t^{3} - 3 t^{2}$ . At $t = 3$ , is it speeding up or slowing down? The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? 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.

A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 72 t + 9$ , $t \geq 0$ . Find the velocity at $t = 2$
A particle moves along a line with $s (t) = - 2 t^{3} + 18 t^{2} - 48 t - 7$ , $t \geq 0$ . Find the acceleration at $t = 4$
A particle moves along a line with $s (t) = 2 t^{3} - 15 t^{2} + 24 t - 6$ , $t \geq 0$ . When is it at rest?
A particle's position is $s (t) = t^{3} - 6 t^{2} + 9 t$ . At $t = 4$ , is it speeding up or slowing down?
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative?
A particle moves along a line with $s (t) = - 4 t^{3} + 30 t^{2} - 72 t + 7$ , $t \geq 0$ . Find the velocity at $t = 2$
A particle moves along a line with $s (t) = - 2 t^{3} + 42 t^{2} - 288 t + 3$ , $t \geq 0$ . Find the acceleration at $t = 9$
A particle moves along a line with $s (t) = - 4 t^{3} + 48 t^{2} - 84 t$ , $t \geq 0$ . When is it at rest?
A particle's position is $s (t) = t^{3} - 6 t^{2}$ . At $t = 5$ , is it speeding up or slowing down?
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest?
A particle moves along a line with $s (t) = - 4 t^{3} + 54 t^{2} - 168 t$ , $t \geq 0$ . Find the velocity at $t = 4$
A particle moves along a line with $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 9$ , $t \geq 0$ . Find the acceleration at $t = 3$
A particle moves along a line with $s (t) = 4 t^{3} - 36 t^{2} + 60 t + 1$ , $t \geq 0$ . When is it at rest?
A particle's position is $s (t) = t^{3} - 9 t^{2} + 24 t$ . At $t = 0$ , is it speeding up or slowing down?
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative?
A particle moves along a line with $s (t) = 2 t^{3} - 27 t^{2} + 108 t$ , $t \geq 0$ . Find the velocity at $t = 9$
A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 144 t + 8$ , $t \geq 0$ . Find the acceleration at $t = 0$
A particle moves along a line with $s (t) = - 2 t^{3} + 33 t^{2} - 168 t - 2$ , $t \geq 0$ . When is it at rest?
A particle's position is $s (t) = t^{3} - 3 t^{2}$ . At $t = 3$ , is it speeding up or slowing down?
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest?



Sigma Prep · sigmaprep.io/worksheets Name Motion Along a LineAnswer keyVersion 1 Date Period Answer each question about the particle. A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 72 t + 9$ , $t \geq 0$ . Find the velocity at $t = 2$ $- 48$ A particle moves along a line with $s (t) = - 2 t^{3} + 18 t^{2} - 48 t - 7$ , $t \geq 0$ . Find the acceleration at $t = 4$ $- 12$ A particle moves along a line with $s (t) = 2 t^{3} - 15 t^{2} + 24 t - 6$ , $t \geq 0$ . When is it at rest? $t = 1, t = 4$ A particle's position is $s (t) = t^{3} - 6 t^{2} + 9 t$ . At $t = 4$ , is it speeding up or slowing down?Speeding up The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? $0 < t < 4.5$ A particle moves along a line with $s (t) = - 4 t^{3} + 30 t^{2} - 72 t + 7$ , $t \geq 0$ . Find the velocity at $t = 2$ $0$ A particle moves along a line with $s (t) = - 2 t^{3} + 42 t^{2} - 288 t + 3$ , $t \geq 0$ . Find the acceleration at $t = 9$ $- 24$ A particle moves along a line with $s (t) = - 4 t^{3} + 48 t^{2} - 84 t$ , $t \geq 0$ . When is it at rest? $t = 1, t = 7$ A particle's position is $s (t) = t^{3} - 6 t^{2}$ . At $t = 5$ , is it speeding up or slowing down?Speeding up The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? $t = 2$ and $t = 5$ A particle moves along a line with $s (t) = - 4 t^{3} + 54 t^{2} - 168 t$ , $t \geq 0$ . Find the velocity at $t = 4$ $72$ A particle moves along a line with $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 9$ , $t \geq 0$ . Find the acceleration at $t = 3$ $36$ A particle moves along a line with $s (t) = 4 t^{3} - 36 t^{2} + 60 t + 1$ , $t \geq 0$ . When is it at rest? $t = 1, t = 5$ A particle's position is $s (t) = t^{3} - 9 t^{2} + 24 t$ . At $t = 0$ , is it speeding up or slowing down?Slowing down The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? $t > 4.5$ A particle moves along a line with $s (t) = 2 t^{3} - 27 t^{2} + 108 t$ , $t \geq 0$ . Find the velocity at $t = 9$ $108$ A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 144 t + 8$ , $t \geq 0$ . Find the acceleration at $t = 0$ $- 84$ A particle moves along a line with $s (t) = - 2 t^{3} + 33 t^{2} - 168 t - 2$ , $t \geq 0$ . When is it at rest? $t = 4, t = 7$ A particle's position is $s (t) = t^{3} - 3 t^{2}$ . At $t = 3$ , is it speeding up or slowing down?Speeding up The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? $t = 3$ and $t = 5$ 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.

A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 72 t + 9$ , $t \geq 0$ . Find the velocity at $t = 2$ $- 48$
A particle moves along a line with $s (t) = - 2 t^{3} + 18 t^{2} - 48 t - 7$ , $t \geq 0$ . Find the acceleration at $t = 4$ $- 12$
A particle moves along a line with $s (t) = 2 t^{3} - 15 t^{2} + 24 t - 6$ , $t \geq 0$ . When is it at rest? $t = 1, t = 4$
A particle's position is $s (t) = t^{3} - 6 t^{2} + 9 t$ . At $t = 4$ , is it speeding up or slowing down?Speeding up
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? $0 < t < 4.5$
A particle moves along a line with $s (t) = - 4 t^{3} + 30 t^{2} - 72 t + 7$ , $t \geq 0$ . Find the velocity at $t = 2$ $0$
A particle moves along a line with $s (t) = - 2 t^{3} + 42 t^{2} - 288 t + 3$ , $t \geq 0$ . Find the acceleration at $t = 9$ $- 24$
A particle moves along a line with $s (t) = - 4 t^{3} + 48 t^{2} - 84 t$ , $t \geq 0$ . When is it at rest? $t = 1, t = 7$
A particle's position is $s (t) = t^{3} - 6 t^{2}$ . At $t = 5$ , is it speeding up or slowing down?Speeding up
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? $t = 2$ and $t = 5$
A particle moves along a line with $s (t) = - 4 t^{3} + 54 t^{2} - 168 t$ , $t \geq 0$ . Find the velocity at $t = 4$ $72$
A particle moves along a line with $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 9$ , $t \geq 0$ . Find the acceleration at $t = 3$ $36$
A particle moves along a line with $s (t) = 4 t^{3} - 36 t^{2} + 60 t + 1$ , $t \geq 0$ . When is it at rest? $t = 1, t = 5$
A particle's position is $s (t) = t^{3} - 9 t^{2} + 24 t$ . At $t = 0$ , is it speeding up or slowing down?Slowing down
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the acceleration negative? $t > 4.5$
A particle moves along a line with $s (t) = 2 t^{3} - 27 t^{2} + 108 t$ , $t \geq 0$ . Find the velocity at $t = 9$ $108$
A particle moves along a line with $s (t) = 4 t^{3} - 42 t^{2} + 144 t + 8$ , $t \geq 0$ . Find the acceleration at $t = 0$ $- 84$
A particle moves along a line with $s (t) = - 2 t^{3} + 33 t^{2} - 168 t - 2$ , $t \geq 0$ . When is it at rest? $t = 4, t = 7$
A particle's position is $s (t) = t^{3} - 3 t^{2}$ . At $t = 3$ , is it speeding up or slowing down?Speeding up
The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? $t = 3$ and $t = 5$

How to do these

$s (t) = t^{2} + 3 t$ . Find the velocity at $t = 2$ as a limit of average velocities

Average velocity from 2 to 2 + h: (s(2 + h) - s(2))/h.
s(2 + h) = 4 + 4h + h² + 6 + 3h and s(2) = 10, so the top is 7h + h².
Divide by h to get 7 + h, then let h go to 0.

The answer is $v (2) = 7$ .

Where students go wrong

Putting h = 0 into the difference quotient before simplifying. That gives 0/0; expand and cancel the h first, then take the limit.

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 $s (t) = 2 t^{2} - 20 t - 8$ , $t \geq 0$ . Find the velocity at $t = 4$ → $- 4$
Acceleration at a moment: A particle moves along a line with $s (t) = 2 t^{2} - 20 t - 8$ , $t \geq 0$ . Find the acceleration at $t = 4$ → $4$
When the particle is at rest: A particle moves along a line with $s (t) = 2 t^{2} - 20 t - 8$ , $t \geq 0$ . When is it at rest? → $t = 5$
Direction, and speeding up or slowing down: A particle's position is $s (t) = t^{3} - 12 t^{2} + 36 t$ . At $t = 7$ , is it moving left or right? → Moving right
Read off a velocity graph: The graph shows $v (t)$ for $0 \leq t \leq 8$ . When is the particle at rest? → $t = 2$ and $t = 6$
v(t) from a graphed s(t), sketched: $s (t) = - t^{3} + 3 t^{2} + 2$ , $0 \leq t \leq 3$ → $v (t) = - 3 t^{2} + 6 t$ ; a parabola through $(0, 0)$ and $(2, 0)$ , vertex $(1, 3)$

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.

Precalculus is usually taken around 11th or 12th grade, 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 6: velocity at a moment, acceleration at a moment, when the particle is at rest, direction, and speeding up or slowing down, read off a velocity graph and v(t) from a graphed s(t), sketched. 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.

Precalculus motion along a line worksheet

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Motion Along a Line

Motion Along a LineAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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