The integral of velocity is displacement: where the particle ended up compared with where it began. The distance it travelled is the integral of the size of the velocity, and the two stop agreeing the moment it turns round.
A particle has velocity v(t)=2t−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 travelled 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 travelled, 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)=6t−24. Find its displacement from t=0 to t=5.→ −45
Total distance travelled
A particle has velocity v(t)=6t−24. How far does it travel from t=0 to t=5?→ 51
Position from velocity
A particle has velocity v(t)=6t−24 and s(0)=0. extFinds(t)ext.→ 3t2−24t
Questions about these worksheets
Yes. Every displacement and distance travelled 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 travelled 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: displacement, total distance travelled and position from velocity. Tick as many as you want and set how many of each, or let it spread them evenly.
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.