Kt, where k is a positive constant. The central force is given by:

F(r) = -k^2re-2kt

To see this, we can use the fact that the radial component of the force is given by:

F(r) = m(d^2r/dt^2) - mr(dθ/dt)^2

Since the motion is purely radial, we have dθ/dt = 0, and so the second term vanishes. We can also express the acceleration in terms of r and its derivatives:

d^2r/dt^2 = (d/dt)(dr/dt) = (dr/dt)(d/dt)(e-kt) = -k(e-kt)(dr/dt)

Plugging this into the expression for the force, we get:

F(r) = -mkr(e-kt)(dr/dt) = -k^2re-2kt

as claimed.

To analyze the motion, we can use conservation of angular momentum to write:

mr^2(dθ/dt) = h

where h is a constant. We can solve for dθ/dt to get:

dθ/dt = h/(mr^2)

Substituting the expression for r, we get:

dθ/dt = h/(me-2kt)

Integrating with respect to t, we get:

θ = (h/km)e2kt + C

where C is an integration constant. This tells us that the particle is spiraling around the origin with increasing speed as time goes on.

To analyze the radial motion, we can use conservation of energy to write:

1/2m(dr/dt)^2 + V(r) = E

where V(r) is the potential energy due to the central force, which we can find by integrating:

V(r) = -∫F(r)dr = k^2/2(e-kt)^2 + C'

where C' is another integration constant. Plugging in the expression for r and rearranging, we get:

(dr/dt)^2 = 2(E - k^2/2(e-kt)^2 - C')/m

Taking the square root and integrating with respect to t, we get:

t = ∫dr/√[2(E - k^2/2(e-kt)^2 - C')/m]

This integral can be evaluated in terms of elliptic functions, but the details are not important for our purposes. The key point is that the radial motion is periodic, with the particle oscillating back and forth between a minimum and maximum radius. The period of the motion depends on the energy E, which determines the amplitude of the oscillation, and the constants k and m, which determine the shape of the spiral.