We've all seen those wondrous videos of melons being tied up with rubber bands and the exploding like a bomb when they have had enough:
The reason this watermelon exploded is because of Elastic Potential Energy (EPE). The rubber bands have this in much greater quality, but the melon has a much lower cap to this.
The second the melon explodes the part of the melons detaching exploding everywhere have lots of kinetic energy.
In this video, the melon explodes with so much force that it knocks the man over. This same principle is how guns work.
A firing pin strikes gun powder in the back casing of the round causing expanding gunpowder gases to push the bullet forwards.
The same mechanism is why guns make a loud crack and why silencers exist. When gases don't have anywhere or not enough room to escape, they will make a loud crack.
Using the concepts of potential energy, we can determine that bottle being pumped up for an explosion has:
- Kinetic potential energy
- Elastic potential energy
- Pneumatic (pressure) potential energy
Using volume and material strength, we can determine a formula to calculate:
- How much energy is needed to make the bottle explode
- How far up the bottle cap will fly under Earth's 1G of gravity after the bottle exploded (9.8 m/s)
- Determine whether it would be possible to make the bottle explode using a hand pump
During an explosion, it's safe to assume somethings:
- The bottle cap will not travel through the air by itself- plastic from the bottle's body will be attached
- The bottle cap will most likely not travel up in a straight line
- The total mass of the bottle will only be determined after the explosion
- Pascal Principles said that the pressure applied to a confined fluid is transmitted undiminished to every part of the fluid and to the walls of the container (see diagrams below)
The force exerted on any specific area of the bottle wall is defined by:
\[ F = P \cdot A \]
Where \(P\) is the internal pressure and \(A\) is the surface area of the section being analyzed.
The work done (\(W\)) to compress the air is calculated using:
\[ W = P_1 V_1 \ln\left(\frac{P_2}{P_1}\right) \]
The total energy (\(E\)) released during an adiabatic expansion at the burst point is:
\[ E = \frac{(P_{burst} - P_{atm}) \cdot V}{\gamma - 1} \]
To find the velocity of the cap, we use the kinetic energy formula:
\[ K_e = \frac{1}{2} m v^2 \]
The maximum height (\(h\)) reached by the cap under gravity (\(g = 9.8 \, \text{m/s}^2\)) is:
\[ h = \frac{v_0^2}{2g} \]
There are some forces at play here which can do very heavy damages to the human body. It's important to stand at least 3 meters back with a plastic shield in front of you for max protection whilst doing this experiment.
For reference here is an Aviation saying:
A person who weighs roughly 70kg will be lifted into the air at 10 hecto-pascals (hpa) over 1 ms^2.
To blow up a plastic bottle, a person would need to exert 150 psi which is 10342 hpa. This would need to be done using a road bike pump.
Children, don't do this experiment. Parents do this experiment with your children supervising you.
Going back to guns- another rule of thumb is that 50,000 psi is released from a high pressure round in a gun. This is ~3,500,000 hpa.
PRESSURE_COMPARISON // Data_Stream
| Scenario | Pressure (PSI) | Pressure (hPa) |
|---|---|---|
| 70kg Mass Lift (@ 1m/s²) | approx. 0.15 | 10 |
| Plastic Bottle Expansion | 150 | 10,342 |
| High Pressure Round (Firearm) | 50,000 | approx. 3,500,000 |
CAP_BALLISTICS_ANALYZER // Data_Stream
CAP BALLISTIC ANALYZER
PHYSICS_DECODED // ARCHIVE_01
The massive calculator and data table above should give a clue about the forces being dealt with here.
1, 187.56 m/s is the speed the cap will leave the bottle after exploding, where this is around 4500 km/h. This is around mach 3 - 4.
This assumes 100% energy efficiency, which is just not possible under standard Earth conditions. Instead this is what happens:
- Gas Expansion Speed: The air itself can only expand at the speed of sound (approx. 343 m/s). The cap cannot be pushed faster than the gas pushing it. Once the cap hits the speed of sound, the air can no longer "keep up" to apply more force.
- The "Pop" Loss: Most of that energy is lost to the atmosphere as a shock-wave (the loud bang) and the rapid expansion of the bottle itself.
- Mass of the Air: You aren't just moving the 1.1g cap; you are also accelerating the mass of the air escaping the bottle, which significantly reduces the energy available for the cap.
From no. 1 above, it's likely the cap would not travel more than 343 m/s, at the speed of sound. This is because the cap cannot be pushed faster than the air pushing it.
Based of equation on the top, it would not be possible to build a explosive bottle to send a cap to the moon. This is because of energy inefficiencies where a lot of the energy is directed onto a center rim 1/2 around the vertical distance of the bottle (despite Pascal's Principle).
Due to the first principle, to send a bottle cap to the moon, we would have to build it outside of Earth's atmosphere where it would cruise at a growing speed running into the moon because of the lack of friction and 0 resistance of space.
Thank you for reading.