I'm in a high school AP physics class but I suspect this question is considerably above the course level because I don't think it was intended to be a part of the project, so I'm not sure which forum to put it in. I would like to find the initial velocity required to launch a ping pong ball 25 meters at x angle (whatever the optimum angle for this would be), accounting for air resistance but assuming no wind. I measured the radius of the sphere (ping pong ball) to be 18.8 mm (.018 m) giving it a cross sectional area of .0111 m^2. The mass is 2.6g. The outside air density is about 1.22 kg/m^3. I came across the equation: F = 1/2*cd*p*a*v^2 where F is the force of air resistance on the object, cd is a coefficient of drag (I guess .5 for sphere?), p is air density, a is the cross section area of the object, and v is velocity. I can see how this is useful in determining the force of air resistance at a specific velocity, but I have no idea where to go from here to calculate initial velocity required to get it to fly a specific distance since the velocity, and consequently force of the air resistance, are dependant on each other and constantly changing. The reason I would like to know is because my physics teacher has designed a competition in which we must build a catapult that launches a ping pong ball to hit a target. Points (translated later into the grade for the project) are based on how accurately you can hit the target. Unfortunately, the target distance is random for each person (lol?.. it gets worse) from 5 meters to 25 meters. Just at a glance I have not the slightest clue how someone aiming for a target 25 meters away is going to compete with someone aiming for a target 5 meters away. However, factor in air resistance on PING PONG balls and the distance disadvantage/advantage is exponential, thus giving someone with a shorter distance a ginormous advantage (much more than 5x in 5 m vs 25 m because it's exponential). Since I'de rather my grade not be decided by lottery, I'de like to have some mathematical evidence to show how ridiculous this competition is. It is worth noting that the catapult cannot exceed w/l/h dimensions of 2'x2'x3' respectively. Assuming putting energy into this mechanism wasn't a problem, it seems even getting to the initial velocity required in 2 feet would require an acceleration so large that it would be near impossible to build something capable of withstanding that force.