# Is the coefficient of restitution negative?

I recently came across a formula for the coefficient of restitution:

Why is there a negative sign?

• Because whoever wrote that doesn't know what they are talking about, most likely. The way they used the "absolute value" signs makes that fairly obvious. Your Wikipedia link has a better definition. – alephzero Oct 20 '18 at 15:47
• @alephzero What is wrong with the absolute value signs? – user190081 Oct 20 '18 at 16:06
• @user190081, physics is not math. The concepts of physics drive the math and not the other way around. The negative sign along with the absolute values in the numerator and denominator indicates an unsophisticated math viewpoint AND especially an unsophisticated physics viewpoint. Whoever developed that equation needs to put quite a bit more time into studying BOTH math AND physics. – David White Oct 21 '18 at 1:19
• I found this equation in a book by Thomas Povey, a professor of engineering science at Oxford. Since he had quoted this formula twice, I would have thought this was deliberate, rather than a mistake. – Jeremy Feng Oct 21 '18 at 10:22

This expanation deals with objects that do not pass through one another.

Firstly replace the bars with brackets to make the formula correct. If you don't, it becomes wrong because of the little negative sign.

The coefficient of restitution is a number with a value that lies in the range of 0 to 1. It can never be negative.

If the formular is presented in that form, the denominator represents the relative "velocity" of approach and the numerator (excluding the negative sign) represents the relative "velocity" of separation.

Now since one object is always chosen as the frame of reference of the other, either of the two (i.e relative velocity of approach/relative velocity of separation) can have a negative sign as a consequence of calculation, not both.

NB: the negative sign i'm refering to in the last statement isn't the one in the equation. It's actually a result of calculating.

The negative sign in the equation is put in order create some sought of a sign balance. Let's say e=1 , equating the denominator to the numerator without the negative sign wouldn't be right since we mentioned that either one is always negative ( due to the approach and separation of an object with our reference frame) so in order to compansate for that we have to plug in a negative sign to the formula at the correct side. Which is the one shown above.

Feel free to correct me, i'm just a learner after all.

Hope this helped a bit.

• @Jeremy: just read your wiki link where they say "e" can be less than zero. I guess my answer applies for 0</=e</=1 only, which is what most textbooks and teachers teach. – Energy Oct 20 '18 at 19:11
• Just in case i didn't answer your question. The coefficient of restitution can be negative or positive depending on the sitution. If we are dealing with situations where momentum is conserved, then "e" can never be negative. But in special situations, "e" can take a negative value (momentum isn't usually cnserved in this case) like the example of the bullet given by your link. – Energy Oct 20 '18 at 19:29