Air Cannon

Projectile Motion and Air Drag

Projectile Motion

When you throw a ball to a friend across a field the path the ball takes through the air can be predicted, and that is what goes through your mind unconsciously when you start the throw. If you throw the ball with more force (producing higher ball velocity) you have to change the angle to still get it to land near your friend.

Drag

As the projectile passes through the air, the air heats up as the air-molecules rub against each other. This energy is taken from the kinetic energy of the projectile, which causes the projectile to slow down. The deceleration can be thought of as being caused by a force on the object in the direction opposite of travel, the drag force. The drag force depends on how fast the object is moving, the shape of the object, and the density of the air; it follows the Drag Equation. Unfortunately, this drag force makes calculating the trajectory much more difficult, and it seems to be impossible to find a closed form solution, meaning it must be solved through iteration.

Calculating the Trajectory

• Ignoring Drag

  • The path a ball takes (ignoring air resistance) through the air is that of a parabola. To make the ball go the furthest (ignoring air resistance) you must throw the ball at a 45 degree angle.

• With Drag

  • Air resistance makes things complicated, and the optimal angle to throw or shoot the object generally decreases.

    Interestingly, guns like those found on battleships may choose angles higher than 45 degrees for maximum range. The distance they fire the projectiles is great enough that they can take advantage of the decreasing density of the air at high altitudes (meaning less drag) which, in that case, extends the range better than the advantage of a shallow angle.

• When To Ignore Drag

  • It is a major pain to calculate trajectories with drag, so is that really necessary? It totally depends on the situation. Note the squared dependance of the velocity on the drag - maximum velocity has the greatest significance on the effects of drag. However, even if the velocities are low if the object is not very dense it is possible for the drag term to still be significant, as the gravitational force would, in turn, be lower. Unfortunately, drag can play a major role in most places where you would actually use the projectile motion equation - hitting baseballs, launching water balloons, etc. Play around with the Projectile Motion Machine on this page to get an idea of the significance of drag and the optimal angle to shoot/throw things in real life.

Optimal Angle to Fire

In most situations the angle to fire the projectile at to get maximum distance is less than 45 degrees. Air drag causes the projectile to slow down quickly, and it turns out to be more advantageous to use the high initial speed for progress in the horizontal direction before the projectile slows down. Although rare, situations exist where firing a projectile at 45 degrees or more may be advantageous because the the difference in the density of the air at extremely high altitudes is enough to cause significantly less drag.

(Wikipedia Article For More Info )

Compressed Gas as Energy Storage

Compressed air can be used as a high power energy storage device, but it has some major disadvantages. A lot of energy can be wasted in the compression and decompression and the tanks required to store the compressed gas add some serious weight.

Viability of Compressed Air Energy Storage [TheOilDrum]

Comparison of Energy Densities [Wikipedia]

Pressure to Force

It is measured in Pascals [Pa, N/m^2]. Pressure is a lot like voltage, in that it measures the potential relative to another reference. There are two references commonly used, one which results in “gauge” pressure reading and another the “absolute” pressure reading. The gauge pressure is the pressure difference between the external environment, and the pressure you are measuring. When you have the gauge disconnected and open to the environment the gauge will read 0. On the other hand, absolute pressure is measured relative to a total vaccum. When the absolute pressure gauge is disconnected it will read atmospheric pressure, or ~101.3 kPa.

Converting pressure to a force is easy, and fundamental when dealing with pressures in confined spaces. Pressure is seen as a force in all directions perpendicular to it’s confinement. The force is the pressure times the area. If you were to look at a potato cannon, the force pushing the potato out of the cannon is the pressure behind the potato times the cross-sectional area of the barrel.

Definition of Pressure [Wikipedia]

Force to Acceleration

Newton’s Second Law of Motion states, “Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum.” Momentum = mv, Force = d (mv) / dt. Hence, the commonly referenced, F = ma.

Acceleration [Wikipedia]

Newton's Laws of Motion [Wikipedia]

Acceleration to find Muzzle Velocity

To get to muzzle velocity, we can integrate the acceleration. The acceleration need not be constant, but we will choose it to be to make the integration simpler, as it is easy to imagine a constant pressure -> constant force -> constant acceleration situation.

a = F/m

v = Ft/m + v_0

Great, now we have velocity dependent on time, but we want it dependant on position, so we have to do more calculations

x = .5 Ft^2/m + v_0 t + x_0

In the situation where the starting velocity is 0, and starting position is 0, we get:

x = .5 Ft^2/m and v = Ft/m

Then we combine the v and x equations to get:

v = (2Fx/m)^.5