Initial Velocity - Projectile Motion

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Discussion Overview

The discussion revolves around calculating the initial velocity of a projectile, specifically a firework, in the context of its vertical and horizontal displacements. Participants explore the equations of motion and the effects of air resistance on these calculations, with a focus on deriving initial velocities based on given time and height parameters.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to derive the initial velocity of a firework based on its time to explosion and height, using specific equations of motion.
  • Another participant suggests that the initial velocity at launch is equivalent to the velocity of the object after falling from its peak altitude, providing a reference for calculations.
  • A participant clarifies that the 600 ft height is the explosion height and suggests that the maximum height may exceed this, indicating a need to calculate both the initial velocity and the final velocity at the time of explosion.
  • One response confirms the validity of the initial velocity equation used but notes that it only accounts for vertical motion and neglects air resistance, stating that the horizontal component cannot be determined from the provided data.
  • A participant expresses interest in incorporating air resistance into their calculations and provides additional known values, indicating a desire for more precise modeling.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the calculations or the effects of air resistance, and multiple competing views regarding the initial velocity and its components remain present throughout the discussion.

Contextual Notes

Limitations include the neglect of air resistance in initial calculations, the dependency on the accuracy of provided height and time values, and the inability to determine the horizontal component of velocity with the given data.

vaio23
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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.
 
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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?
 
vaio23 said:
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.
 
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
 

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