Hello, My end goal is to calculate the vertical and horizontal displacement of an object that explodes in the air e.g. a firework. However, I am having difficulty deriving the initial velocity of the object. I know the approximate time is 5.0 seconds from the ground to explosion and the height is 600 ft (182.88m). So far I have used the equations v(i) = (d / t) - [(a * t) / 2] and V(i) = a*t. Any help would be greatly appreciated.
The initial velocity to fire is the same as the velocity an object would be traveling at after falling from its peak altitude. Formulas and calculator can be found here: http://www.gravitycalc.com/ If the time is 5 seconds, the altitude would be 123 meters and the initial velocity would be 49 meters/second. If the altitude is 183 meters, the time is 6.1 seconds at an initial velocity of 59 meters/second.
The 600 ft (182.88m) is the height in which the fireworks breaks and we do know from industry averages that a shell of this size takes 5.0 seconds to reach this height. It is likely that the maximum height is greater than 600 ft which would mean the shell still has a final velocity at the time of break. Is there a way to calculate for these two unknowns?
The first equation is fine. That will tell you the initial velocity (vertical component only) which will allow the projectile to reach a given height in a given time. Note that this neglects air resistance, which may well be significant. There's no way to determine the horizontal component of velocity from the given data.
Thank you for your response it was a big help. If I wanted to become more precise and incorporate air resistance into my formula how would I do this. It was been a long time since I've done something like this. Known Values: Time = 3.8 Aprox Height = 300 Ft Initial velocity = 140 Ft/s Launch Angle = 75 Degrees Weight of shell = 3.5 lbs Also, the radius of the sphere is 3 inches