How do I find velocity when force is not a constant?

[DBaima22]

how can i find velocity if force is not constant, but a function of distance. I tried integrating the equation for the force, but that doesn't account for the mass of the object.

the given equation for the force is F(x)=2.5-x^2, where x is distance and since F=ma I have
a=(2.5-x^2)/m.

When I tried integrating it I got velocity= -1/3x^3+2.5x which does not account for mass

Is there a quotient rule for integration?

 

[cepheid]

Welcome to PF, DBaima22

how can i find velocity if force is not constant, but a function of distance. I tried integrating the equation for the force, but that doesn't account for the mass of the object.

the given equation for the force is F(x)=2.5-x^2, where x is distance and since F=ma I have
a=(2.5-x^2)/m.

When I tried integrating it I got velocity= -1/3x^3+2.5x which does not account for mass

Is there a quotient rule for integration?

It's completely reasonable to assume that the mass of the object is constant. Therefore, this becomes trivial. You can use the relevant integration property, namely that if g(x) = f(x)/a where a = const. then:

$$\int g(x)\,dx = \int \frac{f(x)}{a}\,dx = \frac{1}{a}\int f(x)\,dx$$

If something is constant (i.e. it has no dependence on the integration variable), then it can be brought outside the integral.

 

[DBaima22]

So, just to be clear, if acceleration is (2.5-x^2)/m. I can write velocity as
v= -1/3x^3+2.5x/mass?

 

[cepheid]

So, just to be clear, if acceleration is (2.5-x^2)/m. I can write velocity as
v= -1/3x^3+2.5x/mass?

Yes. BUT there is another problem with your solution that I only just realized (because I wasn't paying close enough attention before). You CAN'T get velocity by integrating acceleration with respect to position. Velocity is the integral of acceleration with respect to TIME. So you'll have to work a little bit harder to figure out how to get velocity. Hint: the chain rule may be of use here.

 

[asteriatic]

There is not a specific quotient rule for integration, but you can still use the general integration rules to solve for velocity. In this case, you will need to use the Chain Rule to account for the mass in the equation.

First, you can rewrite the equation for acceleration as a function of velocity, a(v)=(2.5-v^2)/m. Then, using the Chain Rule, you can rewrite this as a function of distance, a(x)=(2.5-(dx/dt)^2)/m.

Next, you can integrate this equation with respect to time, which will give you the velocity as a function of time. Then, you can use the relationship between velocity and distance (v=dx/dt) to rewrite the equation in terms of distance.

Alternatively, you can also use numerical methods such as Euler's method or Runge-Kutta method to approximate the velocity at different points along the distance axis. These methods take into account the changing force and mass to calculate the velocity at each point.

Overall, finding velocity when force is not constant and a function of distance requires using integration techniques and/or numerical methods to account for the changing force and mass.