distance (d) traveled. When the surface is flat (initial height of the object is zero), the distance traveled: Thus the maximum distance is obtained if θ is 45 degrees. This distance is: Application of the work energy theorem According to the work-energy theorem the vertical component of velocity is: . These formulae ignore aerodynamic drag and also assume that the landing area is at uniform height 0. Angle of reach The "angle of reach" is the angle (θ) at which a projectile must be launched in order to go a distance d, given the initial velocity v. There are two solutions: (shallow trajectory) and (steep trajectory) Angle θ required to hit coordinate (x, y) To hit a target at range x and altitude y when fired from (0,0) and with initial speed v the required angle(s) of launch θ are: The two roots of the equation correspond to the two possible launch angles, so long as they aren't imaginary, in which case the initial speed is not great enough to reach the point (x,y) selected. This formula allows one to find the angle of launch needed without the restriction of . One can also ask what launch angle allows the lowest possible launch velocity. This occurs when the two solutions above are equal, implying that the quantity under the square root sign is zero. This requires solving a quadratic equation for , and we find This gives If we denote the angle whose tangent is by , then This implies In other words, the launch should be at the angle halfway between the target and Zenith (vector opposite to Gravity) Total Path Length of the Trajectory The length of the parabolic arc traced by a projectile L, given that the height of launch and landing is the same and that there is no air resistance, is given by the formula: where is the initial velocity, is the launch angle and is the acceleration due to gravity as a positive value. The expression can be obtained by evaluating the arc length integral for the height-distance parabola between the bounds initial and final displacements (i.e. between 0 and the horizontal range of the projectile) such that: . Trajectory of a projectile with air resistance Air resistance creates a force that (for symmetric projectiles) is always directed against the direction of motion in the surrounding medium and has a magnitude that depends on