Distance along projectile motion [closed]

I was doing kinematics these days, projectile motion. And I did the basic concepts: range, max height, time of flight, projectile on inclined plane. I am curious to know how we would solve a question if it asks us to find the distance not range traveled by the particle undergoing projectile motion. Is it by finding area under the curve or what? Also what is the average speed then.

closed as off-topic by ACuriousMind♦, Ali, Brandon Enright, Kyle Oman, BMSAug 11 '14 at 21:49

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• While introductory level questions are fine, this is really something you can find in every book on the topic, and any decent website on the topic. Some minimal effort is encouraged before asking a question... – Kyle Oman Aug 11 '14 at 20:49
• @Kyle I think this is a very slippery slope. The criterion involving "minimal effort" is essentially maximally subjective. Also, isn't a non-trivial percentage of the stuff people ask on here in every book on the relevant topic? There are questions people have asked about Lie algebras (as one example) that are very similar in their basic-ness to this question relative to the subject matter and can be found in any book on Lie algebras in physics, but I never see those get put on hold. I'm a bit concerned about where this could lead. – joshphysics Aug 11 '14 at 23:08

For any curve in two dimensions described by a position $(x(t), y(t))$ as a function of time, the speed of such a curve at a time $t$ is \begin{align} |\mathbf v(t)| = \sqrt{\dot x(t)^2 + \dot y(t)^2} \end{align} where overdots here mean derivatives with respect to time $t$. To find the total distance traveled along the curve from a time $t_a$ to a time $t_b$, (namely the length of the curve), one simply integrates this speed with respect to time; \begin{align} \ell_{t_b, t_a} = \int_{t_a}^{t_b} |\mathbf v(t)|\, dt = \int_{t_a}^{t_b}\sqrt{\dot x(t)^2 + \dot y(t)^2}\, dt. \end{align} For projectile motion, $x(t)$ and $y(t)$ take a specific form, and I'll leave it to you to perform the integrals. The average speed is then defined as the total distance traveled divided by the total time of travel; \begin{align} \text{average speed from $t_a$ to $t_b$} = \frac{\ell_{t_b,t_a}}{t_b-t_a}. \end{align}