Water Rocket - Rate at which water exits

In summary, the conversation was about a homework question regarding a water rocket and its thrust, and the student needed help setting it up using algebra. They provided equations and attempted a solution, but were struggling. Another user gave them direction on how to solve the problem and the student was able to work through it and get the correct answer.
  • #1
NMS09
2
0

Homework Statement



We're currently studying Momentum & Collisions and Work and Energy

Use algebra – no Calculus

I’d appreciate help in setting it up. I seem to be missing one equation since I keep coming up with two unknown variables.

A water rocket (2-liter half filled with water and pressurized with compressed air) can develop a thrust of 300 N.

Question: At what rate would water have to come out of the rocket to develop that thrust?

The jet of water emerging from the bottle has a diameter of 2.2 cm. (.022 m.).
The water density is 1,000 kg/cu. m.

The answer provided is 10.7 kg/s.


Homework Equations



Equations we’ve been using recently:

Momentum: p = mv

Change of momentum: F = delta p / delta time

Force: F = ma

Force & Impulse: F = mvf - mvi / delta time

Conservation of Momentum: m1v1,i + m2v2,i = m1v1,f + m1v1,f + m2v2,f Note: Those are subscripts after v1 , ve, etc. I need to learn how to use the formulas on this site.



The Attempt at a Solution



I calculated:

The area of the exit hole = 1.52 x 10^-3 m^2
Water in the rocket = 1 kg

I then tried the convservaton of momentum equation but could not get the 300 N into a form to use on one side of the equation. Everything I tried seem to need a "delta t" value, which I don't know and was not given.

The free body diagram has the forward thrust of the rocket created by the rearward explusion of the presurized water. The 300 N must overcome Earth's gravity (-9.81 m / sec^2).

Sorry for not being more organized, but I've spent 3 or 4 hours going around on this one.
 
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  • #2
The force depends on the speed of the water being ejected.

Assume that the water comes out at a speed v relative to the container. The rocket thrust is, as you have stated:

[tex]F = \Delta p/\Delta t = v\Delta m/\Delta t[/tex]

where [itex]\Delta m[/itex] is the mass of a volume of water ejected in time [itex]\Delta t[/itex].

Since the water is being ejected at the speed v, [itex]\Delta m[/itex] is [itex]\rho \dV = \rho A\Delta s = \rho Av\Delta t[/itex]. So the rate of mass expulsion is [itex]\Delta m/\Delta t = \rho Av[/itex]

Work out the expression for force and then find the speed that is needed to reach 300 N. From that you can calculate the rate at which mass has to be ejected.

AM
 
  • #3
Andrew --

Thank you for your time and that direction. It helped a lot and I was able to work through the problem to get the correct answer.

The steps made sense and helped me understand what was going on.
 

1. How does the rate at which water exits affect the performance of a water rocket?

The rate at which water exits a water rocket plays a crucial role in its performance. This is because the force generated by the water leaving the rocket determines the thrust, which propels the rocket forward. A higher rate of water exiting will result in a greater thrust and therefore a higher velocity.

2. What factors affect the rate at which water exits a water rocket?

There are several factors that can affect the rate at which water exits a water rocket. These include the pressure inside the rocket, the size and shape of the nozzle, the amount of water in the rocket, and the angle at which the rocket is launched.

3. Can the rate at which water exits a water rocket be controlled?

Yes, the rate at which water exits a water rocket can be controlled to some extent. This can be achieved by adjusting the pressure inside the rocket, changing the size or shape of the nozzle, or altering the amount of water in the rocket. However, external factors such as wind and air resistance can also affect the rate of water exit.

4. What is the optimal rate at which water should exit a water rocket?

The optimal rate at which water should exit a water rocket depends on various factors, such as the design of the rocket and the desired trajectory. Generally, a higher rate of water exit will result in a greater thrust and therefore a higher velocity. However, if the rate is too high, it can cause instability and decrease the overall performance of the rocket.

5. How can the rate at which water exits be measured?

The rate at which water exits a water rocket can be measured using various methods, such as timing the flight of the rocket and calculating the average velocity, or using a flow rate sensor attached to the nozzle. Another way to estimate the rate of water exit is by observing the height and duration of the water column ejected from the rocket.

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