Solving for Constant Centripetal Acceleration: Understanding Spiral Motion

[Ale98]

Assume an object accelerating at a certain value dV/dt. If this object was traveling in a circular motion the centripetal force would increase as the object moves faster.
To maintain centripetal acceleration constant while the object is accelerating (in its forward motion dV/dt) I think it would need to follow some sort of spiral path so that after a certain time the object would be traveling faster however the radius also increases.

Does anybody know what type of spiral would keep centripetal acceleration constant? What formula would this spiral have?

Thanks in advance for the help.

 

[DrStupid]

The differential equation should be

$\dot v = a_t \cdot \frac{v}{{\left| v \right|}} + a_r \cdot \frac{{n \times v}}{{\left| {n \times v} \right|}}$

but I'm too lazy to solve it.

edit: equation amended for a spiral in a plane with the normal n

 

[Ale98]

In your equation is ˙v=at

Just to make sure i understood your notation

 

[DrStupid]

The acceleration $\dot v$ is the sum of tangential $a_t \cdot \frac{v}{{\left| v \right|}}$ (= your dV/dt) and the radial acceleration $a_r \cdot \frac{{n \times v}}{{\left| {n \times v} \right|}}$ (= centripetal acceleration).

 

[Ale98]

Okay, but I am confused on how I can find the formula for a spiral to keep centripetal acceleration constant after I solve the integral.

 

[DrStupid]

Okay, but I am confused on how I can find the formula for a spiral to keep centripetal acceleration constant after I solve the integral.

Just keep $a_r$ constant.