Instantaneous Speed & Resistance: Prove Magnitude Proportional to Cube

[homestar2004]

A particle moving along the x-axis is acted upon by a resisting force which is such that at a time t for it to travel a distance x is given by t=Ax^2+Bx+C, where A, B, and C are constants. show that the magnitude of the resisting force is proportional to the cube of the instantaneous speed.


I know the chain rule needs to be used and it is just a calculus problem, but I can't seem to start it.

 

[jdstokes]

A particle moving along the x-axis is acted upon by a resisting force which is such that at a time t for it to travel a distance x is given by t=Ax^2+Bx+C, where A, B, and C are constants. show that the magnitude of the resisting force is proportional to the cube of the instantaneous speed.


I know the chain rule needs to be used and it is just a calculus problem, but I can't seem to start it.

Start by writing down $dt /dx = \cdots$

 

[homestar2004]

dt/dx= 2Ax+Bx

Now I know I need to get velocity, which is dx/dt, but the only way I can see is to multiply the above by dx/dt, and I don't know if that is right. What about the distance x? Where does it come in?

 

[Hootenanny]

dt/dx= 2Ax+Bx

Now I know I need to get velocity, which is dx/dt, but the only way I can see is to multiply the above by dx/dt, and I don't know if that is right. What about the distance x? Where does it come in?

HINT:

$$\frac{dx}{dt} = \frac{1}{\left(dt/dx\right)}$$

You might also want to recheck your derivative.