# Curved motion, r(φ) when r is accelerating

1. Jan 28, 2015

### kamil.borkowski

Hi there!

I have a problem to find an equation of r(t), which will help me to describe motion of blue dot. I have seen many cases where there is a linear motion and you can write φ=ωt , r=V0t, but here there is an acceleration. I think I can write φ=εt2/2, but what about r(t) ? I have tried differencial equation
r = at2/2 + r0 → r = (r''-rφ')t2/2 + r0, but not sure if it's right. I need all this to have equation of path r(φ). r0 is initial position of blue dot, where V0=0 and ω=0. Cheers!

Last edited: Jan 28, 2015
2. Jan 28, 2015

### Stephen Tashi

There are many possible formulas for r(t) that would produce a curved trajectory. To pose a specific mathematical problem, you have to state a complete set of requirements for r(t). Is this a physics problem? Are forces involved?

3. Jan 28, 2015

### kamil.borkowski

I am trying to describe a motion of a clay target in a spring-powered target thrower. You can involve some forces (f.ex. friction between target and friction tunnel of an arm), but this complicates case even more. The only constant I can assume is velocity when target leaves the arm (27m/s). Everything else has to be calculated. The main thing is to have ε of this arm, because then from M=I⋅ε we can have required force in a spring.

4. Jan 28, 2015

### Stephen Tashi

The instantaneous velocity in polar coordinates is given by the vector $\frac{dr}{dt}r$ in the radial direction and $\frac{d\theta}{dt} \theta$ in the tangential direction. If you want the magnitude of this velocity vector to be 27 m/sec at the time t when the target reaches the end of the arm, you have to keep in mind the contribution of the component $\frac{dr}{dt} r$ to the velocity. It will be important to have a physically reasonable formula for $r(t)$.

5. Jan 28, 2015

### Staff: Mentor

Without forces, how would you know how the object moves?
There are many possible motions that end with a speed of 27m/s.

6. Jan 29, 2015

### A.T.

I think you at least need to also know the time it needs to get to that speed (from rest?), to get the (constant?) angular acceleration of the thrower. Then I would use the rotating reference frame of the thrower to compute r(t) using the inertial forces there.