A particle has velocity v(t) = -18t^2 + 126t - 180 and s(0) = -7. Find s(t).

s(t) = -6t^3 + 63t^2 - 180t - 7

a(t) = 4t + 4, v(0) = 4, s(0) = 2. Find v(t) and s(4).

v(t) = 2t^2 + 4t + 4; s(4) = 278/3

a(t) = 4t + 6, v(0) = -2, s(0) = -1. Find v(t) and s(2).

v(t) = 2t^2 + 6t - 2; s(2) = 37/3

a(t) = 2t - 5, v(0) = 6, s(0) = 6. Find v(t) and s(3).

v(t) = t^2 - 5t + 6; s(3) = 21/2

A particle has a(t) = 6t - 4, v(0) = 3, and s(0) = -2. Find v(t) and s(t).

v(t) = 3t^2 - 4t + 3; s(t) = t^3 - 2t^2 + 3t - 2

A particle has a(t) = 6, v(0) = 3, and s(0) = -4. Find its position, velocity, speed and acceleration at t = 4.

s = 56; v = 27; Speed 27; a = 6

Worksheets · Calculus

Displacement and distance traveled worksheet

The integral of velocity is displacement: where the particle ended up compared with where it began. The distance it traveled is the integral of the size of the velocity, and the two stop agreeing the moment it turns round.

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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 Displacement and Distance Date Period Answer each question about the particle. Give exact values. Find displacement from velocity A particle has velocity $v (t) = 18 t^{2} - 90 t + 72$ . Find its displacement from $t = 0$ to $t = 7$ . A particle has velocity $v (t) = - 6 t^{2} + 48 t - 90$ . Find its displacement from $t = 0$ to $t = 6$ . A particle has velocity $v (t) = 18 t^{2} - 126 t + 108$ . Find its displacement from $t = 0$ to $t = 8$ . Find total distance traveled A particle has velocity $v (t) = - 18 t^{2} + 198 t - 540$ . How far does it travel from $t = 2$ to $t = 8$ ? A particle has velocity $v (t) = 18 t^{2} - 180 t + 432$ . How far does it travel from $t = 3$ to $t = 8$ ? A particle has velocity $v (t) = - 12 t^{2} + 84 t - 144$ . How far does it travel from $t = 1$ to $t = 5$ ? Find position from velocity A particle has velocity $v (t) = 12 t^{2} - 36 t + 24$ and $s (0) = - 4$ . Find $s (t)$ . A particle has velocity $v (t) = - 18 t^{2} + 126 t - 180$ and $s (0) = - 7$ . Find $s (t)$ . A particle has velocity $v (t) = 12 t^{2} - 48 t + 36$ and $s (0) = 7$ . Find $s (t)$ . Integrate a velocity graph A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 8$ . Find the displacement from $t = 0$ to $t = 4$ . A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 7$ . Find the total distance traveled from $t = 0$ to $t = 6$ . A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 9$ . Find the total distance traveled from $t = 0$ to $t = 9$ . Find velocity and position from acceleration $a (t) = 4 t + 4$ , $v (0) = 4$ , $s (0) = 2$ . Find $v (t)$ and $s (4)$ . $a (t) = 4 t + 6$ , $v (0) = - 2$ , $s (0) = - 1$ . Find $v (t)$ and $s (2)$ . $a (t) = 2 t - 5$ , $v (0) = 6$ , $s (0) = 6$ . Find $v (t)$ and $s (3)$ . Find displacement and distance for trig velocity $v (t) = 7 \sin t$ on $[0, \frac{3 π}{2}]$ . Find the displacement and the total distance. $v (t) = 2 \sin t$ on $[0, 2 π]$ . Find the displacement and the total distance. $v (t) = \sin t$ on $[0, π]$ . Find the displacement and the total distance. Work back from acceleration A particle has $a (t) = 6 t - 4$ , $v (0) = 3$ , and $s (0) = - 2$ . Find $v (t)$ and $s (t)$ . A particle has $a (t) = 6$ , $v (0) = 3$ , and $s (0) = - 4$ . Find its position, velocity, speed and acceleration at $t = 4$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Displacement and Distance

Date Period

Answer each question about the particle. Give exact values.

Find displacement from velocity
A particle has velocity $v (t) = 18 t^{2} - 90 t + 72$ . Find its displacement from $t = 0$ to $t = 7$ .
A particle has velocity $v (t) = - 6 t^{2} + 48 t - 90$ . Find its displacement from $t = 0$ to $t = 6$ .
A particle has velocity $v (t) = 18 t^{2} - 126 t + 108$ . Find its displacement from $t = 0$ to $t = 8$ .

Find total distance traveled
A particle has velocity $v (t) = - 18 t^{2} + 198 t - 540$ . How far does it travel from $t = 2$ to $t = 8$ ?
A particle has velocity $v (t) = 18 t^{2} - 180 t + 432$ . How far does it travel from $t = 3$ to $t = 8$ ?
A particle has velocity $v (t) = - 12 t^{2} + 84 t - 144$ . How far does it travel from $t = 1$ to $t = 5$ ?

Find position from velocity
A particle has velocity $v (t) = 12 t^{2} - 36 t + 24$ and $s (0) = - 4$ . Find $s (t)$ .
A particle has velocity $v (t) = - 18 t^{2} + 126 t - 180$ and $s (0) = - 7$ . Find $s (t)$ .
A particle has velocity $v (t) = 12 t^{2} - 48 t + 36$ and $s (0) = 7$ . Find $s (t)$ .

Integrate a velocity graph
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 8$ . Find the displacement from $t = 0$ to $t = 4$ .
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 7$ . Find the total distance traveled from $t = 0$ to $t = 6$ .
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 9$ . Find the total distance traveled from $t = 0$ to $t = 9$ .

Find velocity and position from acceleration
$a (t) = 4 t + 4$ , $v (0) = 4$ , $s (0) = 2$ . Find $v (t)$ and $s (4)$ .
$a (t) = 4 t + 6$ , $v (0) = - 2$ , $s (0) = - 1$ . Find $v (t)$ and $s (2)$ .
$a (t) = 2 t - 5$ , $v (0) = 6$ , $s (0) = 6$ . Find $v (t)$ and $s (3)$ .

Find displacement and distance for trig velocity
$v (t) = 7 \sin t$ on $[0, \frac{3 π}{2}]$ . Find the displacement and the total distance.
$v (t) = 2 \sin t$ on $[0, 2 π]$ . Find the displacement and the total distance.
$v (t) = \sin t$ on $[0, π]$ . Find the displacement and the total distance.

Work back from acceleration
A particle has $a (t) = 6 t - 4$ , $v (0) = 3$ , and $s (0) = - 2$ . Find $v (t)$ and $s (t)$ .
A particle has $a (t) = 6$ , $v (0) = 3$ , and $s (0) = - 4$ . Find its position, velocity, speed and acceleration at $t = 4$ .



Sigma Prep · sigmaprep.io/worksheets Name Displacement and DistanceAnswer keyVersion 1 Date Period Answer each question about the particle. Give exact values. Find displacement from velocity A particle has velocity $v (t) = 18 t^{2} - 90 t + 72$ . Find its displacement from $t = 0$ to $t = 7$ . $357$ A particle has velocity $v (t) = - 6 t^{2} + 48 t - 90$ . Find its displacement from $t = 0$ to $t = 6$ . $- 108$ A particle has velocity $v (t) = 18 t^{2} - 126 t + 108$ . Find its displacement from $t = 0$ to $t = 8$ . $- 96$ Find total distance traveled A particle has velocity $v (t) = - 18 t^{2} + 198 t - 540$ . How far does it travel from $t = 2$ to $t = 8$ ? $330$ A particle has velocity $v (t) = 18 t^{2} - 180 t + 432$ . How far does it travel from $t = 3$ to $t = 8$ ? $168$ A particle has velocity $v (t) = - 12 t^{2} + 84 t - 144$ . How far does it travel from $t = 1$ to $t = 5$ ? $68$ Find position from velocity A particle has velocity $v (t) = 12 t^{2} - 36 t + 24$ and $s (0) = - 4$ . Find $s (t)$ . $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 4$ A particle has velocity $v (t) = - 18 t^{2} + 126 t - 180$ and $s (0) = - 7$ . Find $s (t)$ . $s (t) = - 6 t^{3} + 63 t^{2} - 180 t - 7$ A particle has velocity $v (t) = 12 t^{2} - 48 t + 36$ and $s (0) = 7$ . Find $s (t)$ . $s (t) = 4 t^{3} - 24 t^{2} + 36 t + 7$ Integrate a velocity graph A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 8$ . Find the displacement from $t = 0$ to $t = 4$ . $10$ A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 7$ . Find the total distance traveled from $t = 0$ to $t = 6$ . $6$ A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 9$ . Find the total distance traveled from $t = 0$ to $t = 9$ . $\frac{25}{2}$ Find velocity and position from acceleration $a (t) = 4 t + 4$ , $v (0) = 4$ , $s (0) = 2$ . Find $v (t)$ and $s (4)$ . $v (t) = 2 t^{2} + 4 t + 4$ ; $s (4) = \frac{278}{3}$ $a (t) = 4 t + 6$ , $v (0) = - 2$ , $s (0) = - 1$ . Find $v (t)$ and $s (2)$ . $v (t) = 2 t^{2} + 6 t - 2$ ; $s (2) = \frac{37}{3}$ $a (t) = 2 t - 5$ , $v (0) = 6$ , $s (0) = 6$ . Find $v (t)$ and $s (3)$ . $v (t) = t^{2} - 5 t + 6$ ; $s (3) = \frac{21}{2}$ Find displacement and distance for trig velocity $v (t) = 7 \sin t$ on $[0, \frac{3 π}{2}]$ . Find the displacement and the total distance.displacement $7$ ; distance $21$ $v (t) = 2 \sin t$ on $[0, 2 π]$ . Find the displacement and the total distance.displacement $0$ ; distance $8$ $v (t) = \sin t$ on $[0, π]$ . Find the displacement and the total distance.displacement $2$ ; distance $2$ Work back from acceleration A particle has $a (t) = 6 t - 4$ , $v (0) = 3$ , and $s (0) = - 2$ . Find $v (t)$ and $s (t)$ . $\begin{array}{l} v (t) = 3 t^{2} - 4 t + 3 \\ s (t) = t^{3} - 2 t^{2} + 3 t - 2 \end{array}$ A particle has $a (t) = 6$ , $v (0) = 3$ , and $s (0) = - 4$ . Find its position, velocity, speed and acceleration at $t = 4$ . $\begin{array}{l} s = 56 \\ v = 27 \\ Speed 27 \\ a = 6 \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Displacement and DistanceAnswer keyVersion 1

Date Period

Answer each question about the particle. Give exact values.

Find displacement from velocity
A particle has velocity $v (t) = 18 t^{2} - 90 t + 72$ . Find its displacement from $t = 0$ to $t = 7$ . $357$
A particle has velocity $v (t) = - 6 t^{2} + 48 t - 90$ . Find its displacement from $t = 0$ to $t = 6$ . $- 108$
A particle has velocity $v (t) = 18 t^{2} - 126 t + 108$ . Find its displacement from $t = 0$ to $t = 8$ . $- 96$

Find total distance traveled
A particle has velocity $v (t) = - 18 t^{2} + 198 t - 540$ . How far does it travel from $t = 2$ to $t = 8$ ? $330$
A particle has velocity $v (t) = 18 t^{2} - 180 t + 432$ . How far does it travel from $t = 3$ to $t = 8$ ? $168$
A particle has velocity $v (t) = - 12 t^{2} + 84 t - 144$ . How far does it travel from $t = 1$ to $t = 5$ ? $68$

Find position from velocity
A particle has velocity $v (t) = 12 t^{2} - 36 t + 24$ and $s (0) = - 4$ . Find $s (t)$ . $s (t) = 4 t^{3} - 18 t^{2} + 24 t - 4$
A particle has velocity $v (t) = - 18 t^{2} + 126 t - 180$ and $s (0) = - 7$ . Find $s (t)$ . $s (t) = - 6 t^{3} + 63 t^{2} - 180 t - 7$
A particle has velocity $v (t) = 12 t^{2} - 48 t + 36$ and $s (0) = 7$ . Find $s (t)$ . $s (t) = 4 t^{3} - 24 t^{2} + 36 t + 7$

Integrate a velocity graph
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 8$ . Find the displacement from $t = 0$ to $t = 4$ . $10$
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 7$ . Find the total distance traveled from $t = 0$ to $t = 6$ . $6$
A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 9$ . Find the total distance traveled from $t = 0$ to $t = 9$ . $\frac{25}{2}$

Find velocity and position from acceleration
$a (t) = 4 t + 4$ , $v (0) = 4$ , $s (0) = 2$ . Find $v (t)$ and $s (4)$ . $v (t) = 2 t^{2} + 4 t + 4$ ; $s (4) = \frac{278}{3}$
$a (t) = 4 t + 6$ , $v (0) = - 2$ , $s (0) = - 1$ . Find $v (t)$ and $s (2)$ . $v (t) = 2 t^{2} + 6 t - 2$ ; $s (2) = \frac{37}{3}$
$a (t) = 2 t - 5$ , $v (0) = 6$ , $s (0) = 6$ . Find $v (t)$ and $s (3)$ . $v (t) = t^{2} - 5 t + 6$ ; $s (3) = \frac{21}{2}$

Find displacement and distance for trig velocity
$v (t) = 7 \sin t$ on $[0, \frac{3 π}{2}]$ . Find the displacement and the total distance.displacement $7$ ; distance $21$
$v (t) = 2 \sin t$ on $[0, 2 π]$ . Find the displacement and the total distance.displacement $0$ ; distance $8$
$v (t) = \sin t$ on $[0, π]$ . Find the displacement and the total distance.displacement $2$ ; distance $2$

Work back from acceleration
A particle has $a (t) = 6 t - 4$ , $v (0) = 3$ , and $s (0) = - 2$ . Find $v (t)$ and $s (t)$ . $\begin{array}{l} v (t) = 3 t^{2} - 4 t + 3 \\ s (t) = t^{3} - 2 t^{2} + 3 t - 2 \end{array}$
A particle has $a (t) = 6$ , $v (0) = 3$ , and $s (0) = - 4$ . Find its position, velocity, speed and acceleration at $t = 4$ . $\begin{array}{l} s = 56 \\ v = 27 \\ Speed 27 \\ a = 6 \end{array}$

How to do these

A particle has velocity $v (t) = 2 t - 4$ . How far does it travel from $t = 0$ to $t = 3$ ?

The velocity is zero at t equals 2, so it turns round there.
From 0 to 2 it moves 4 one way, and from 2 to 3 it moves 1 back.
So the distance traveled is 5, while the displacement is only 3 the other way.

The answer is $5$ .

Where students go wrong

Integrating the velocity straight through when the particle turns round. Past the turn the velocity is negative and that stretch subtracts, which gives the displacement rather than the distance. Split at every time the velocity is zero.

Also called displacement, total distance traveled, position from velocity, motion along a line revisited or integral of velocity.

What you can put on this worksheet

Displacement: A particle has velocity $v (t) = - 4 t + 24$ . Find its displacement from $t = 0$ to $t = 8$ . → $64$
Total distance traveled: A particle has velocity $v (t) = 6 t - 24$ . How far does it travel from $t = 2$ to $t = 6$ ? → $24$
Position from velocity: A particle has velocity $v (t) = 6 t - 24$ and $s (0) = - 5$ . Find $s (t)$ . → $s (t) = 3 t^{2} - 24 t - 5$
From a velocity graph: A particle moves along a line with the velocity $v (t)$ shown, for $0 \leq t \leq 9$ . Find the displacement from $t = 6$ to $t = 9$ . → $- \frac{3}{2}$
From acceleration, with initial values: $a (t) = 1$ , $v (0) = 0$ , $s (0) = 1$ . Find $v (t)$ and $s (1)$ . → $v (t) = t$ ; $s (1) = \frac{3}{2}$
Trig velocity: $v (t) = 3 \sin t$ on $[0, 2 π]$ . Find the displacement and the total distance. → displacement $0$ ; distance $12$
From acceleration: functions, a moment, distance: A particle has $a (t) = 5$ , $v (0) = - 7$ , and $s (0) = 0$ . Find its position, velocity, speed and acceleration at $t = 3$ . → $\begin{array}{l} s = \frac{3}{2} \\ v = 8 \\ Speed 8 \\ a = 5 \end{array}$

Questions about these worksheets

Yes. Every displacement and distance traveled 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 displacement and distance traveled 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 7: displacement, total distance traveled, position from velocity, from a velocity graph, from acceleration, with initial values, trig velocity and from acceleration: functions, a moment, distance. Tick as many as you want and set how many of each, or let it spread them evenly.

Displacement and distance traveled worksheet

Select your difficulty

Find displacement from velocity

Find total distance traveled

Find position from velocity

Integrate a velocity graph

Find velocity and position from acceleration

Find displacement and distance for trig velocity

Work back from acceleration

Find displacement from velocity

Find total distance traveled

Find position from velocity

Integrate a velocity graph

Find velocity and position from acceleration

Find displacement and distance for trig velocity

Work back from acceleration

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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