To help simply the problem, consider a coordinate system that is attached to the satellite and moves with it. This will eliminates the satellite's orbital motion and it allows the problem to be solved like it is initially at rest.

Also, the coordinate system should be rotated so that the impact point is on the y-axis. This will simplify the equations and calculations.

Meteoroid Impact Causing
Angular Momentum

Linear Momentum Before
and After Impact

Since the satellite is not initially spinning and is at rest, the total system linear momentum (both particle and satellite) is only in the meteoroid, or

     L_m = m_mv_m

Assume that after the impact, the meteoroid becomes imbedded in the satellite. The linear momentum is

     L_s+m = (m_s + m_m) v_s

Since there are no external forces acting on the system, linear momentum must be conserved, or

     L_m = L_s+m

Solving for the linear velocity of the satellite gives

If the assumption is made that m_m<< m_s, then this reduces to

         = (0.33 kg-km/s)/(100 kg) = 3.3 m/s

Angular Momentum Before
and After Impact

Just prior to impact, the angular momentum about the satellite center of the whole system (meteoroid and satellite) is

     H_c1 = - r m_m v_m cos45

This assumes that the particle momentum is zero or very small and has no effect on the overall angular momentum. Since after the impact the meteoroid is imbedded in the satellite, the angular momentum is

     H_c2 = I_s ω_s

There are no other external moments in the system, so angular momentum must be conserved giving

     H_c2 = H_c1

Substituting for H_c1 and H_c2, gives the rotational or angular velocity of the satellite as

Firing the thrusters applies a pure torque to the satellite; therefore, the linear momentum and hence the linear velocity of the satellite remains unchanged,

Before and After Thruster Fired

At this point, the satellite is spinning due to the particle impact and the thruster need to fire to stop this rotation. This requires the thruster to apply a moment impulse on the satellite.

From the Principle of Impulse and Momentum, the integral of the applied moments about the center of rotation equals the change in angular momentum,

The only applied moments to the system are the thrusters firing opposite the rotation:

Integrating gives

     2 r F_r t = -ω_s2 I_s

Substituting for the angular velocity ω_s gives,

     2 r F_r t = -[-(r p_m cos45) / I_s] I_s

Notice, both I_s and r terms cancel. Simplifying gives the required thruster firing-time to stop the rotation,

It is interesting to note that the firing time does not depend on the satellite radius, r, or its moment of inertia, I_s.