Suppose that the motion time of the projectile is t and the acceleration of gravity is g, which is constant. Regardless of air resistance, according to Newton's law, there are:
x=vtcosa①,y=vtsina-( 1/2)gt^2②
① and ② are the equations of motion of the projectile.
1. Let y=0, and the range can be obtained. According to formula ②, the exercise time at this time is t=2Vsina/g③, and the range when formula ③ is substituted into formula ① is:
X=V^2sin2a/g④
It can be seen from formula (4) that the range x is proportional to the square of the initial velocity v of the projectile and to the sine value of the angle of twice. When the initial velocity is constant, sin2a= 1, that is, A = 45, the maximum range is V2/g.
Second, when the muzzle height from the ground is h, ② becomes:
y=H+Vtsina-( 1/2)gt^2⑤
Let y=0, and the range can be obtained. According to formula ⑤, the time at this time is: t = (vsina+√ ((vsina) 2+2gh))/g. If this formula is substituted into ①, the range is:
x=v^2sin2a/(2g)+vcosa√((vsina)^2+2gh)/g⑥
It can be seen from formula ⑥ that under the same shooting angle, the higher the muzzle is from the ground, the farther the range is. When H=0, Formula 6 becomes Formula 6.
In addition, it is quite complicated to calculate the best shooting angle strictly by Formula 6, and approximate results are generally used in practice. When h is small, the optimal shooting angle is approximately given by the following formula:
sina≈( 1-gH/(2V^2))/√2⑦
The maximum range is roughly given by the following formula:
X≈V^2/g+H⑧
Third, the actual ballistic equation is much more complicated than the above, and there are many factors that affect the range, not just the shooting angle. Not only the influence of air resistance should be considered, but also it is related to meteorological conditions, altitude, projectiles, artillery and so on. How to improve the hit accuracy is an important topic in ballistic theory.