Science Buddies: "Ask an Expert"

So in my class we are making baking soda and vinegar rockets. I'm a huge fan of rockets and orbital mechanics but I don't know how to apply the rocket propulsion equation to this case becuase it is more of an explosion with the cork popping our of a bottle than a constant burn. My question is how would you find the force exerted when the cork blows out of the bottle from the pressure built inside?

One way that comes to mind is to remember that force=mass*acceleration, or F=ma. You can measure the mass of the cork, and if you have a video camera that can record in slow motion (many inexpensive snapshot camera can do this, albeit in low resolution), you can hold the bottle in place and record the cork being shot out. Place a ruler or some marked scale behind the bottle and cork so that it's in the frame. Since you know the frame rate of the camera, you can see how far it went in successive frames, calculate the acceleration and from that you can calculate the force.

Howard

Power5000,

Thank you for your question. Rocket propulsion works on Newton's 3rd law: for every action there is an equal and opposite reaction. Even though your fuel (baking soda and vinegar) explodes out in what seems like all at once that mass being ejected imparts a force in the opposite direction to your rocket. That being said you can apply the thrust equation to your rocket as well but you will need to make a couple of assumptions or measurements if you can. below is a series of calculations that I've been successful in estimating the performance of a bottle rocket. I've used it for a water/air rocket, but I've modified for baking soda and vinegar for fuel. I've made some assumptions for some of the parameters like the size (volume) of your bottle and such, but I've tried to point these out where I've made them. here it goes:

1) Assuming a 2 liter bottle = volume of bottle = Vbottle = 2L

2) Acceleration due to gravity = g = 9.8m/s^2

3) Assuming an internal pressure at liftoff = Pi = 40psi (this is an assumption, you should try to measure the max pressure the cork will hold before it pops off)

4) Select solution volume: 700 mL = 0.7L (assumption based on roughly 33% fill of solution to total bottle volume)

5) Find mass of liquid solution
a) M = Mass = density x volume (m=rho(vol))
b) Vol = 700 ml = 700 cm^3 = 0.0007 m^3
c) rho = density of solution = 1050kg/m^3 (this is assumed based on some online searching on the density of vinegar but not verified)
d) M = 1050 kg/m^3 x 0.0007 m^3 = 0.735 kg

6) Calculate average water mass flow rate, mdot, of solution out of nozzle: mdot = A x Cd x (sqrt(2(rho)Pave)
a) Find area of nozzle = A = (pi)r^2
b) pi = 3.14159
c) diameter of nozzle = d = 2.5cm = 0.025 m
d) radius of nozzle = r = d/2 = 0.025m/2 = 0.0125m
e) A = 3.14159(0.0125^2) = 0.0004909m^2
f) drag constant of fuel exiting nozzle assumed = Cd = 0.98
g) average pressure = Pave = (Pi + Pf)/2 or (Pi(1+Vi/Vf))/2, since PiVi = PfVf, so Pf = (PiVi)/Vf
h) initial pressure = Pi = 40psi
i) initial volume of air = vol of bottle – vol of solution = Vi_air = 2L – 0.7L = 1.3L
j) final volume = Vf = 2L
k) final pressure = Pf = 40(1.3)/2 = 26psi
l) Pave = (40 + 26)/2 = 33psi
m) Convert psi to N/m^2; 1psi = 6,894.76N/m^2
n) 33 psi = 227,527.08 N/m^2; Note N = kg(m)/s^2
o) Mass flow rate of solution = mdot = 0.0004909m^2 x 0.98 x (sqrt(2 x 1050kg/m^3 x 227527.08 N/m^2) = 10.5159kg/s

7) Calculate solution exit velocity and Thrust
a) Solution exit velocity = V = mdot/rho(A) = 10.5159kg/s / (1050kg/m^3)(0.0004909m^2) = 20.4015m/s
b) Rocket thrust = T = mdot x V = 10.5159kg/s x 20.4015m/s = 214.54kgm/s^2 = 214.54N

This is the answer to your question; to keep going and find out how far your rocket should fly then:

8 ) Net force on Rocket; where force of drag = fd = 0 (this assumes no drag; but we'll account for it in a later step)
a) Net force = F = T – fd – (mavg x g)
b) F = 214.54N – 0 – ((0.3kg + 0.7 kg)/2 x (9.8m/s^2) = 209.64kgm/s^2 = 209.64N

9) Calculate average mass of rocket = Mave = mass of empty rocket + mass of solution selected
a) Mass of rocket empty = Mempty = 300g = 0.3kg (you'll need to weigh your rocket without fuel)
b) Mass of solution = Mfuel = 0.7kg

10) Calculate Rocket acceleration as a result of net force acting on mass
a) F = Mave x a, so a = F/Mave
b) acceleration = a = (209.64N)/((0.3 + 0.7 kg)/2 = 419.28m/s^2

11) Range can be calculated if you know the amount of time it takes for the solution to exit and the velocity of the rocket
a) Time to expel solution = t = M/mdot = 0.7kg / 10.5159kg/s = 0.0665s
b) Initial Velocity of rocket = Vrocket = a x t = 419.28m/s^2 x 0.0665s = 27.91m/s
c) Range is V^2rocket x sin 2theta / g
d) Launch angle = Theta = 45deg
e) Range = R = (27.91m/s)^2 x sin 2(45) / 9.8 m^2 = 79.49m

12) Estimating final range affected by drag
a) Coefficient of drag = Dc = 0.2 (Dc is some number between 0 and 1; I selected a low value for a bottle rocket with nose cone)
b) Drag force D = 1 - Dc = 1 - 0.2 = 0.8
c) Estimated range with drag = Rdrag = R x D = 79.49m x 0.8 = 63.59m

You can use this algorithm to calculate expected results and then test against reality for a neat science project.

I'd like to know how closely your results match predicted.