# Significant Figures in long physics calculations?

## Main Question or Discussion Point

Hi all,

Newbie here. Forgive if I'm starting this thread in the wrong place! Started college physics this semester and although the physics is not the most challenging, I'm running into lots of trouble with the sig figs! First of all, when doing long calculations, where do you round the sig figs? Just a quick problem: Plane in the wind, heading south at 185 km/hr, after an hour the place has only gone 135 km at 15 degrees East of South. I know how to do this, but I ran into trouble with the first part in finding the Southern displacement.

cos(15)*135 = southern displacement.

Does this equation equate to 2 significant figures because of cos(15)? Or is it 3 from 135?

Another example that I had trouble with is a question about finding where a long jumper will land. I found the initial velocity, used that to find how long the jumper was in the air with Vi, and solved for t with the quadratic equation. But, again, the sig figs are haunting me. Should I round them when I find the initial velocity and then use the rounded velocity in the quadratic equation? Or should I use the raw initial velocity in the quadratic equation, then round after? I am thoroughly confused with this, because then I don't know where in the quad formula to round!

If someone could clear this up I would REALLY appreciate this. I am normally an A+ student but my grades are suffering because of the sig fig confusion. Again, I thank you so much if you can help even guide me in the right direction.

(EDIT) I am not asking for any of my homework to be solved. I can complete it, just the sig figs part I don't know where to start!

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
My general suggestion is to keep everything symbolic and only plug in numbers at the very end. Never round things off in the middle of a computation.

In the end, significant digits are a bit of a poor man's error propagation and I do not think teachers should be too strict with applying arbitrary rules on how many significant digits should be included as long as it is reasonable. For example, the numbers 1.3 and 9.4 imply very different relative errors even though they have the same number of significant digits.

jtbell
Mentor
I agree with Orodruin. Ideally, you should get a single equation that gives you the final answer, then plug all your given numbers into that, calculate the answer in one fell swoop on your calculator, and round off that final answer.

If the algebra is too cumbersome for that method, you can break up the calculation into separate steps. Or you may be given a problem that's already divided up into separate steps that you have to give the intermediate answers for (parts a, b, c, etc.) In that case, you should leave the answer for each step in your calculator to use as input for the next step, unrounded. If you need to give that number as an answer to a sub-part of the problem, round it off when writing it down, but leave the unrounded version in the calculator.

SteamKing
Staff Emeritus