Did someone say “Rocket Science”?!

Okay, now that you’ve tried your hand at launching a few water rockets, what makes them go? And better yet, how can we make them go even better?!

The forces involved in launching a water rocket.

First we need to look at a basic water rocket and understand what forces are involved. Immediately after launch, the forces acting on the rocket are an upward thrust which is a reaction to the water being expelled through the nozzle, and then, acting against this are the force of gravity, and drag forces. The force of gravity continually pulls the rocket toward the earth and the strength of this force depends upon the mass (or how much physical material, primarily water) is in the rocket. Drag has two components, one is called parasitic drag and is the result of air rubbing against the surface of the rocket, and the other is called form drag and this depends upon the shape of the rocket (is the rocket fat and bulky, or is it thin and sleek).

Now, how do we figure out how much thrust a water rocket has? Since we’re working with water, lets investigate this problem using a branch of physics called hydrodynamics which is a study of fluid motion. Now, we’ll make some assumptions and simplifications to make our job a little easier. First, we know that moving water experiences drag just like our rocket does while moving through the air. We’re going to neglect this effect for the moment with the understanding that the estimates we get using our calculations are theoretical, maximum values. In practice, our rockets may approach these limits, but they’ll never reach them because of these inherent frictional losses. Another thing we’ll assume is that our water rocket is essentially a cylindrical bottle with an opening, or orifice at its bottom. We won’t worry about the curvature around the bottle’s neck or that it may not be perfectly round. Now, what variables do we have at our disposal that we can control? The first one that comes to mind is the mass, m of the rocket. This is essentially the amount of water we have in the bottle at the time of launch. The mass is also affected by the materials we use to build the rocket body. The next variable might be the pressure, P_o, with which we charge the bottle. We could also change the density of the water, ρ, by mixing some other material in it such as salt or sugar (this would increase the density). And finally, we could restrict the bottle’s opening, a, by fitting it with a bottle cap drilled with a smaller hole.

Now, the first question might be, how fast is the water coming out of the bottle when it’s first launched? In 1644, the Italian physicist, Toricelli, first discovered that the velocity of flow from an opening in a vessel containing water, is the same as that acquired by a body falling from rest in a vacuum through a height equal to the height of water above the opening, or:

Toricelli’s theorem

Where V is the velocity, g is the acceleration due to gravity, and h is the height of the fluid above the orifice. If the vessel is pressurized, as is the case with our water rocket, this additional pressure can be thought of as a column of water with a height sufficient to create this pressure. To do this, imagine a column of water with an area A and a height h. The volume of this column is its area times its height. The mass of water contained in this column is the density of water, ρ, times the volume. And finally the force of weight applied by this column of water is it’s mass multiplied by the gravitational constant, g. Putting this all together we get:

solving for the term gh leaves us with:

This can be substituted for gh in equation 1 to obtain:

Combining equations 1 and 2 produces the general formula for determining the velocity of the water jet leaving the orifice of a pressurized vessel:

where h is the height of the water in the bottle, and g is the gravitational constant of 9.8m/sec².

However, for pressures, P_o, greater than 15psi, the first term, 2gh, can be neglected since it's contribution is less than 1% for a bottle half full of water and the practical equation for us reverts to equation 2, which excludes the term 2gh.

Okay, so now we know how fast the water is moving. So what?

Aren’t we really interested in knowing how much thrust our rocket will have? Yes! And we can find this by considering the equations for impulse and momentum. As shown

Here, F is force, t is time, m is mass, and v is velocity. Solving for the force F (thrust), yields

Mass of water exiting per unit time

The term, m/t, is the mass of fluid per unit time exiting the orifice and V is the velocity of the fluid exiting the orifice, which we just discovered is equal to the square root of 2Po/ρ. The mass leaving the orifice can be found from geometry. Imagine a cylinder of fluid exiting an orifice. The mass contained in that cylinder is equal to it's volume multiplied by the density of the fluid. The volume is equal to the area of the orifice times the velocity of the jet multiplied by the unit time. Mathematically, this is represented as: Solving this for m/t and substituting this into the equation for force yields:

Technically, this is the force that is exerted on an object that the moving water strikes, say for example, the launch pad. The reaction to this force is the upward thrust on the bottle and it’s contents. If we substitute our earlier equation for the water’s velocity into this equation, we get:

Notice that the thrust depends only upon the area of the orifice and the pressure within the bottle. Fluid density does not make an appearance here, and so has no effect on the thrust. The area of the orifice (the neck of the bottle) can only be decreased by any practical methods which would result in a lower thrust. This leaves us with pressure as the only practical parameter to increase the rocket’s thrust. The remaining questions, which will be addressed later, concern the amount of water in the bottle, which affects the duration of the thrust, and aerodynamic drag on the rocket. Both of these factors influence the maximum altitude of the rocket.