Use seperation of varibles to Find velocity interms of time

In summary, the problem involves finding the velocity of a mass m at time t=0, given a drag force f(v)=-cv^3/2 along the x-axis. The solution involves using separation of variables and integrating from 0 to t on one side and 0 to v on the other. The time at which the mass will stop can be found by setting v=0.
  • #1
leonne
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Homework Statement


A mass m has a velocity Vo at time t=0 and coasters along the x-axis were the drag force is f(v)=-cv ^3/2 find v in terms of t and the other given parameter. at what time will it stop.


Homework Equations


F(v)=m dv/dt


The Attempt at a Solution


So i started by using separation of var and got dv= f(v)dt/m then i just take the integral of both sides Is this correct? then i just do it from 0 to t to one side and 0 to v to the other? How would i then find what time it would stop? set v = 0 or something?

Thanks
 
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  • #2
Correct on all accounts. Just make sure that anything with v in it is on the left side and anything with t in it is on the right before you integrate.
 

FAQ: Use seperation of varibles to Find velocity interms of time

How does separation of variables help find velocity in terms of time?

Separation of variables is a mathematical technique used to solve differential equations, which are often used to describe physical systems and their behaviors. By separating the variables of an equation, we can express the relationship between different quantities, such as velocity and time, in a simpler form that can be solved more easily.

What is the process for using separation of variables to find velocity in terms of time?

The process involves isolating the variables on different sides of the equation and then integrating both sides with respect to the appropriate variable. This will result in an expression that relates the two variables, in this case velocity and time. From there, we can solve for velocity in terms of time by manipulating the equation and incorporating any initial conditions or constants.

When is it appropriate to use separation of variables to find velocity in terms of time?

Separation of variables is typically used for solving differential equations that have a linear or separable form. This means that the equation can be written as a linear combination of the two variables, or that the variables can be separated on different sides of the equation. If the equation does not have this form, other techniques may be more suitable for finding the relationship between velocity and time.

What are the limitations of using separation of variables to find velocity in terms of time?

While separation of variables is a powerful technique for solving certain types of differential equations, it is not always applicable. Some equations may not have a separable form, making it impossible to use this method. Additionally, even when separation of variables can be used, it may not always result in a closed-form solution, which means that the relationship between velocity and time cannot be expressed in a simple equation.

Are there any real-world applications of using separation of variables to find velocity in terms of time?

Yes, there are many real-world applications of separation of variables in physics and engineering. For example, it can be used to model the motion of objects in free fall, the diffusion of substances through a medium, or the oscillations of a pendulum. It is a widely used technique for solving differential equations in various fields of science and technology.

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