Modeling with Differential Equations

In summary, the conversation discusses using Newton's second law to find the differential equation for velocity of a falling body with air resistance. The net force is given by Fnet = mg - kv, and the differential equation format is p(x) and q(x). The conversation also mentions a video about first order differential equations.
  • #1
aaronfue
122
0

Homework Statement



A falling body of mass m, encounters air resistance proportional to its instantaneous velocity, v. Use Newton's second law to find the DE for the velocity, v at time, t.

Homework Equations


I've been reading this section quite a bit and am still not getting it! I'd appreciate a simple breakdown of this problem.


The Attempt at a Solution



I know Newton's second law is F=ma, and a=[itex]\frac{dv}{dt}[/itex]= -g, in this problem. I've checked the answer in the back of the book to see if I can work it backwards and then maybe get an idea of what's going on, but I don't think I've dealt with air resistance, k, in any problems before. Even in my physics class.
 
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  • #2
aaronfue said:

Homework Statement



A falling body of mass m, encounters air resistance proportional to its instantaneous velocity, v. Use Newton's second law to find the DE for the velocity, v at time, t.

Homework Equations


I've been reading this section quite a bit and am still not getting it! I'd appreciate a simple breakdown of this problem.

The Attempt at a Solution



I know Newton's second law is F=ma, and a=[itex]\frac{dv}{dt}[/itex]= -g, in this problem. I've checked the answer in the back of the book to see if I can work it backwards and then maybe get an idea of what's going on, but I don't think I've dealt with air resistance, k, in any problems before. Even in my physics class.
Newton's 2nd Law: The net force on a body of mass, m, is given by Fnet = ma .

For this problem Fnet = mg - kv , and is in the downward direction.
 
  • #3
SammyS said:
Newton's 2nd Law: The net force on a body of mass, m, is given by Fnet = ma .

For this problem Fnet = mg - kv , and is in the downward direction.

I see. There was an image next to the question with those being opposite. Now to put ity into a differential equation form? I was watching a video about differential equations and there being a 1st ODE format using p(x) and q(x)?
 
  • #4
aaronfue said:
Now to put it into a differential equation form?
What differential equation relates acceleration to velocity?
 

FAQ: Modeling with Differential Equations

What is the purpose of modeling with differential equations?

Differential equations are mathematical tools used to describe the behavior of systems that change over time. Modeling with differential equations allows scientists to study and understand complex systems by representing them using mathematical equations.

What types of systems can be modeled with differential equations?

Many natural phenomena can be modeled using differential equations, such as population growth, chemical reactions, and fluid dynamics. They can also be applied to artificial systems, such as economic models and engineering systems.

How do you solve a differential equation?

The process of solving a differential equation involves finding a function that satisfies the equation. This can be done analytically, using mathematical methods, or numerically, by approximating the solution using numerical techniques.

What are the limitations of modeling with differential equations?

While differential equations are powerful tools for modeling complex systems, they do have limitations. For example, they may not accurately represent all aspects of a system, and the solutions obtained may only apply to specific conditions and assumptions.

How are differential equations used in real-world applications?

Differential equations are used in a wide range of fields, including physics, biology, economics, and engineering, to study and understand complex systems. They are also used to make predictions and inform decision-making in areas such as weather forecasting, drug development, and financial markets.

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