Differential equation for a rocket

In summary, the problem involves finding the velocity of a rocket traveling through interstellar dust without gravity, with the given equations of k as a constant and drag equal to -bv. The solution involves solving a separable differential equation and using the hint of having m/m0 to a power in the final equation. Dividing and multiplying by certain terms will lead to a solvable equation.
  • #1
DontDeimos
4
1

Homework Statement


A rocket through interstellar dust, no gravity. Solve for v with respect to mass.

gif.latex?%5Cdot%7Bm%7D%3D%5Cfrac%7Bdm%7D%7Bdt%7D%3Dk.gif

k is a constant.
Drag = -bv

Homework Equations


[/B]
gif.latex?m%5Cfrac%7Bdv%7D%7Bdt%7D%3D-%5Cdot%7Bm%7Du-bv.gif


The Attempt at a Solution



To start:

%7Bdv%7D%7Bdm%7D%3D-ku-bv%20%5Crightarrow%20m%5Cfrac%7Bdv%7D%7Bdm%7D%3D-u-%5Cfrac%7Bb%7D%7Bk%7Dv.gif


It should become a separable differential equation, but I am having a lot of trouble solving it for v(m). A hint that was given, is in the final equation you should have has m/m0 to a power.
 
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  • #2
You could divide by ## -u - \frac{b}{k} v## and also by ##m## and then multiply by ##dm## to get

$$\frac{dv}{-u - \frac{b}{k} v} = \frac{dm}{m}$$

This is solvable, see if it gets you the answer you need--I feel like both terms' logarithmic solutions would cancel but because of the coefficients perhaps you will get the correct dependence when you solve for it.
 
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FAQ: Differential equation for a rocket

What is a differential equation for a rocket?

A differential equation for a rocket is a mathematical equation that describes the relationship between the rocket's motion and the forces acting on it. It takes into account factors such as gravity, thrust, and air resistance to predict the rocket's trajectory.

Why is a differential equation important for understanding rocket motion?

A differential equation is important for understanding rocket motion because it allows us to accurately model and predict the behavior of a rocket. By taking into account all the forces acting on the rocket, we can determine its path and make necessary adjustments for a successful launch.

How is a differential equation for a rocket different from a regular equation?

A differential equation for a rocket is different from a regular equation because it involves the rate of change of a variable over time. This allows us to track the changing position, velocity, and acceleration of the rocket as it moves through space.

What factors are included in a differential equation for a rocket?

A differential equation for a rocket includes factors such as the rocket's mass, thrust, air resistance, and gravitational force. These factors all play a role in determining the rocket's trajectory and must be accounted for in the equation.

How is a differential equation for a rocket used in real-world applications?

A differential equation for a rocket is used in real-world applications to design and launch rockets, as well as to predict and control their flight paths. It also has applications in fields such as aerospace engineering and astrophysics.

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