8-3 James Bond is skiing and is being pursued by Goldfinger, also on skis. Assume no friction. Mr. Bond, who has a mass of 100 kg, fires backward a 40 g bullet at 800 m/s. Goldfinger, at 120 kg, fires forward at Bond with a similar weapon. What is the relative velocity change after the exchange of 6 shots each? No bullets hit Bond or Goldfinger (because they are both main characters and it is early in the movie).

Solution: The problem is analyzed with conservation of momentum. The momentum (m_bv_b lets call it) of the bullet fired by Bond increases his momentum by m_BD v_B. We are using little b to stand for the bullet and big B to stand for Bond. The key here is to understand that whatever momentum the bullet has in one direction, Bond has in the exact opposite direction. Remember that each of the 6 bullets Bond fires increases his velocity. Set

 and solve for

.

 or

 The 40 ´ 10^-3 kg bullet is small compared to the 100 kg of Bond and it would not be necessary to subtract that mass from Bond’s mass in the calculation. The D v_B notation is used to indicate that each bullet fired by Bond causes a change in his velocity.

Goldfinger, on the other hand, has his momentum decreased. In his case,

.

 Putting in the numbers

or

Bond goes faster and Goldfinger goes slower with the total change in velocity 0.58 m/s for each pair of shots fired. For six shots, this amounts to a difference of 3.48 m/s. If Bond and Goldfinger had been travelling at the same speeds, then after this exchange Bond would have a relative speed advantage of 3.48 m/s with which to make his escape.

Keep in mind this application of conservation of momentum when doing rocket propulsion problems. The gases expelled from a rocket have a certain momentum in one direction, and the rocket has an equivalent change in momentum in the opposite direction.