Editing Projectile motion (section)

=== Total Path Length of the Trajectory ===
The length of the parabolic arc traced by a projectile, <var>L</var>, given that the height of launch and landing is the same (there is no air resistance), is given by the formula:

:<math>L = \frac{v_0^2}{2g} \left( 2\sin\theta + \cos^2\theta\cdot\ln \frac{1 + \sin\theta}{1 - \sin\theta} \right) = \frac{v_0^2}{g} \left( \sin\theta + \cos^2\theta\cdot\tanh^{-1}(\sin\theta) \right)</math>

where <math>v_0</math> is the initial velocity, <math>\theta</math> is the launch angle and <math>g</math> is the acceleration due to gravity as a positive value. The expression can be obtained by evaluating the [[Arc length|arc length integral]] for the height-distance parabola between the bounds ''initial'' and ''final'' displacement (i.e. between 0 and the horizontal range of the projectile) such that:

:<math>L = \int_{0}^{\mathrm{range}} \sqrt{1 + \left ( \frac{\mathrm{d}y}{\mathrm{d}x} \right )^2}\,\mathrm{d}x = \int_{0}^{v_0^2 \sin(2\theta)/g} \sqrt{1+\left (\tan\theta -{g\over {v_0^2 \cos^2\theta}}x\right)^2}\,\mathrm{d}x .</math>

If the time-of-flight is ''t'',
:<math>L = \int_{0}^{t} \sqrt{v_x^2 + v_y^2}\,\mathrm{d}t = \int_{0}^{2v_0\sin\theta/g} \sqrt{(gt)^2-2gv_0\sin\theta t+v_0^2}\,\mathrm{d}t.</math>