What is the maximum height a sphere shot from cannon can reach?

Answer 1

That depends on basically three things: the angle at which the cannon is fired, the velocity of the projectile, and the acceleration of gravity. The maximum range of the cannon is achieved with a firing angle of 45 degrees to the horizontal. But the maximum height is achieved when the firing angle is 90 degrees to the horizontal, that is, when the cannon is pointing straight up into the air. (Which can be very dangerous to the artillery personnel firing the guns!) So, if we assume that the cannon is on Earth, not some other planet, and fired straight up into the air, can we determine how high the cannonball will go? Yes, if we know the velocity of the cannonball as it leaves the end of the cannon's tube. Once the cannonball leaves the end of the tube, it begins to slow down because of the acceleration of gravity. We can use the energy equations to calculate the maximum height of the cannonball. We know that kinetic energy is defined by the equation Ek = mv2/2. We also know that potential energy (due to altitude) is defined by the equation Ep = mgh. Equating the two, we get mv2/2 = mgh. Rearranging the terms to solve for h, we get: h = v2/2g. (Note that g = 9.8 m/s2 = 32.2 ft/s2.) So, let's say the cannon has a muzzle velocity of 1000 meters per second. The cannonball, therefore, has an initial velocity of 1000 m/s before it starts to slow down. Plugging 1000 into the equation above and solving for h, we calculate the theoretical maximum height as 51,020 meters. In practice, however, the cannonball will not achieve anywhere near that height because of air resistance, which has a tremendous effect at such high speeds.