Rocket with Variable Mass and Air Resistance.

In summary: For example, if you're trying to find the velocity at the bottom of the rocket, compare it to the velocity at the top. Then, use the simpler equation to solve for the bottom velocity.
  • #1
holdenthefuries
5
0
Problem:

A rocket is launched vertically upward. The rocket has a mass Mr and carries Mo of fuel. The fuel burns at a constant rate (ß) and leaves the rocket at speed Ve relative to the rocket. Assume constant gravity (9.8m/sec^2). There is an air resistance given by F(a) = -kV.

Where V is the velocity of the rocket.

Take:

k = .1 n-sec/m
ß = 100 kg/sec
Mr = 1000 kg
Mo = 10000 kg
Ve = 3000 m/sec

What is the terminal velocity of the rocket (the velocity when fuel runs out)?

-----The general solution for this problem is an extension of conservation of momentum for a system of variable mass.

mdv/dt = ∑F(ext) + Vrel(dM/dt)

I have no problem when the external force is solely gravity, however, when this problem presents linear air resistance, I am running into a bit of trouble with the integration.

My setup of the differential equation is as follows:Set dm/dt to constant. dM/dt = ß = 100 kg/sec.

(=)

mdv/dt = 100Ve - mg - kV

--->

dv/dt + kV/m = (100Ve/m - g)


I am attempting to integrate the velocity from V(0) to V(Terminal), and the time from t=0 to t=t(final)= Mo/ß = 10000/100 = 100 seconds.

My question is regarding the velocity portion of the differential equation.

I can't seem to get that V over to the left side of the integral by itself... and I'm wondering if I could get some sort of example on how to do that type of integration, or if I've gone astray somewhere.

Does this integration qualify as a first order differential?

i.e.

dy/dx + P(x)y = Q(x) ?Addendum:

Okay, so I've set up the equation:

m/k(100Ve/m-g) - (m/k(100Ve/m-g)e^(-kt/m))

Thanks!

Sean
 
Last edited:
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  • #2
Yeah, it does. The solution's [tex]ye^{\int p(x)}=\int Q(x)e^{\int P(x)}dx[/tex]
 
  • #3
Tips for making this easier:

Remember that ultimately the mass is constant (since the M comes from the force being applied) and don't try and trick yourself into thinking that it is a variable in the end.

In a rocket formula with linear air resistance, it is helpful to compare your answer to something you'd expect without air resistance.
 

1. What is a Rocket with Variable Mass and Air Resistance?

A Rocket with Variable Mass and Air Resistance is a type of rocket that takes into account the changing mass of the rocket as it expels fuel and the effects of air resistance on its flight. This type of rocket is designed to be more efficient and accurate in its flight trajectory.

2. How does a Rocket with Variable Mass and Air Resistance work?

A Rocket with Variable Mass and Air Resistance works by continuously adjusting its mass as it expels fuel, ensuring that its thrust-to-mass ratio remains constant. It also takes into account the effects of air resistance, using fins and other aerodynamic features to minimize drag and maintain stability during flight.

3. What are the benefits of using a Rocket with Variable Mass and Air Resistance?

The main benefit of using a Rocket with Variable Mass and Air Resistance is increased efficiency and accuracy in its flight. By adjusting its mass and accounting for air resistance, this type of rocket can reach higher speeds and altitudes while maintaining stability. This makes it ideal for space exploration and other long-distance missions.

4. How is a Rocket with Variable Mass and Air Resistance different from a regular rocket?

A Rocket with Variable Mass and Air Resistance differs from a regular rocket in its design and capabilities. While a regular rocket has a fixed mass and does not take into account air resistance, a Rocket with Variable Mass and Air Resistance is able to dynamically adjust its mass and incorporate aerodynamic features to improve its flight trajectory.

5. What are some real-world applications of a Rocket with Variable Mass and Air Resistance?

A Rocket with Variable Mass and Air Resistance has many real-world applications, including space exploration, satellite launches, and military missile systems. It is also being studied for potential use in commercial airliners and other forms of transportation that require efficient and accurate flight.

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