# Using Derivatives and Integrals to Find Velocity: Am I doing this right?

• jumbogala
In summary, the conversation discusses finding the velocity of a function, F(x), given displacement x and constants A and B. The initial attempt is to use F = ma, but then it is realized that a = dv/dt, not dv/dx. The correct approach is to use a = dv/dt = (dx/dt)*(dv/dx) = v(dv/dx) and integrate accordingly. The final conclusion is that the second reasoning is correct and the integration is straightforward.
jumbogala

## Homework Statement

The function given to me is F(x) = A + Bx.

x is the displacement, F(x) is the force as a function of that displacement, and A and B are constants.

From the function, I'm supposed to find the velocity of the function as a function of x.

We also know that the items which follow this function have mass m.

## The Attempt at a Solution

First I tried using F = ma, giving ma = A + Bx. Then I divided by m on both sides to get a = A/m + (B/m)x.

Then I integrated both sides, to get v = whatever the integral of the RHS is.

But then I realized that a = dv/dt, NOT dv/dx. So integrating with respect to dt doesn't work.

So instead I think I should use a = dv/dt = (dx/dt)*(dv/dx) = v(dv/dx)

And work that into my equation somehow. But I'm having a mental debate with myself about whether I was right the first time... can anyone help?

You are right. Proceed.

So my 2nd reasoning is correct, not my first, right?

If I do that it works fine, it's just a simple integration. Thanks!

## 1. How do I use derivatives and integrals to find velocity?

To find velocity using derivatives, you need to take the derivative of the position function with respect to time. This will give you the velocity function. To find velocity using integrals, you need to first find the acceleration function and then integrate it with respect to time.

## 2. Can I use derivatives and integrals to find velocity for any type of motion?

Yes, derivatives and integrals can be used to find velocity for any type of motion, as long as you have a function that describes the position of the object as a function of time.

## 3. How accurate are the results when using derivatives and integrals to find velocity?

The accuracy of the results depends on the accuracy of the initial position, velocity, and acceleration values. If these values are accurate, then the results obtained using derivatives and integrals will also be accurate.

## 4. Can I use derivatives and integrals to find velocity for non-uniform motion?

Yes, derivatives and integrals can be used to find velocity for non-uniform motion. However, the calculations may be more complex as the acceleration may not be constant.

## 5. Are there any limitations to using derivatives and integrals to find velocity?

One limitation is that the initial position, velocity, and acceleration values must be known. If these values are not known, then derivatives and integrals cannot be used to find velocity. Additionally, the calculations may become more complex for non-uniform motion.

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