Assume that: m is the mass of the dart and M is the mass of the wheel, R is the radius of the wheel and v0 is the velocity of the dart just before it strikes and sticks into the edge of the wheel. Assume that the wheel starts at rest. Further assume that the rotation axis of the wheel is the z axis and the direction of flight of the dart (before the collision) was along a line in the xy plane described by the equation y = R. Then this problem can be solved by invoking conservation of angular momentum about the z axis.
The initial angular momentum is Li = m v0 R
The final angular momentum is Lf = I ω where ω is the angular speed of the wheel after the collision (rad/s) and I (the rotational inertia) is given by
I = ½ M R2 + m R2 = (½ M + m) R2
The first term in I is the rotational inertial of disk ( modeling the wheel as a disk of mass M and radius R).
The second term in I is the contribution from the dart sticking into the wheel at the edge of the wheel.
Setting Li = Lf an equation for ω results. It is:
ω = ( v0 /R) [m/(½M + m) ]
The initial kinetic energy is KEi = ½ m v02.
The final kinetic energy is KEf = ½ I ω2 = ½ m v02 [ m/(½ M + m) ]
So the ratio KEf / KEi = m/(½ M + m) which is a value less then unity- which is expected because this is an inelastic collision.