Maximum Range With Air Resistance


I have a question regarding projectiles. I know that a 45 degree angle produces the maxium distance if air restistance is neglected. Is the angle for maxium distance the same, less, or greater than 45 degrees if air restistance is accounted for.


Air resistance for a non-rotating spherical object affects both horizontal and vertical components of the object's trajectory in the same way. The force opposing the motion is proportional to the square of the speed. Since gravity brings the speed in the vertical direction to zero at some point in the flight, the horizontal average speed tends to be faster than the vertical and air resistance therefore has a larger effect on the horizontal motion. The result of this is that the trajectory is no longer symmetrical about its highest point, as it was in the no air resistance case. The descending portion of the trajectory will be cover less ground then the ascending part, making the angle at which the object hits the ground greater than the angle of the shot. The maximum range will occur when the sum of the firing angle and the impact angle are 90 degrees so the firing angle must be reduced from 45 to achieve maximum range.

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