For an ideal same-height launch, resolve ux=ucosθ and uy=usinθ. With upward positive, vy=uy− gt and y=uyt− ½gt2. At the apex vy=0, so tpeak =guy and H=(2g)uy2. Total time is g2uy and range is ux times that time.
The combined range formula is R=gu2sin(2θ). At fixed speed, complementary angles have equal ranges, and 45∘ gives maximum range only for equal launch and landing heights without drag. If the landing level differs, substitute its actual signed vertical displacement into the motion equation rather than using the symmetric flight time.
Example 1
Launch at 20.0ms−1 and 30.0∘, landing at the same height. Use g=9.80ms−2.
uy=10.0ms−1; T=g2uy=2.04s. ux=17.3ms−1; R=35.3m.
Example 2
A launch has uy=14.0ms−1. Find the rise above launch with g=9.80ms−2.
At the top vy=0. From vy2=uy2−2gH, H=2×9.8014.02=10.0m.
Watch out: That result belongs to the fixed-speed, same-height, no-drag model; changed heights, drag or speed can change the optimum.