OK here is my nutty design problem. I want to make a centrifugal nut cracker to crack hazelnuts. The basic design premise is very simple when the nuts are thrown perpendicular to a surface at a desired speed the shells split very nicely into two pieces. Anyways I want to make a centrifugal accelerator that throws/ejects the nuts onto a hard surface. The basic design is a rotating disk with two radially mounted blades. The nuts fall in the center of the disk and are accelerated outward until they are ejected with some velocity V that has both a radial and tangential component (see attached picture). I desire to make the nuts come out with as much radial velocity as possible and as little tangential component as possible. To do this I need to derive the equations for the nuts velocity. I am for now not considering frictional forces or any forces and just trying to get the kinematic relationship. The way I have the blades mounted the tangential velocity is simply v_t=R*W. Where R=radius and W=rotational speed. The radial velocity is where I am stuck. I know centrifugal acceleration is a_r=W^2*R. The problem is that the this acceleration varies with respect to radial position. So the nuts experience a greater and greater acceleration force as they travel along the blade. It is just like being on a merry go round where as you travel to the outside a larger and larger force is pulling out. Anyways I don't know how to treat this equation I guess it could be written as a second order diff. eq like d^2r/dt^2=W^2*R, which if this is the right equation I don't know how to solve, any ideas?