Initial Velocity - Projectile Motion

[vaio23]

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.

 

[TumblingDice]

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.

 

[vaio23]

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?

 

[Doc Al]

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 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.

 

[vaio23]

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