Momentum problem

Question

A 105.9kg archer is standing on level ice (neglect friction) fires a 106.4g arrow to the right at 97.9m/s what is the recoil speed of the archer? (Report your answer to at least the thousandth’s place).

Answer 1

This is a great, simple problem in basic, classical physics. The principle to keep in mind is that the sum of all momentums in a closed systems does not change. If we consider the archer and his arrows to be "the system" and these start at rest, then they have an initial momentum of zero. But then, he fires an arrow away from him. This arrow has a momentum of mass x velocity:

106.4g x 97.9m/s = P_arrow

Let's convert the mass to kilogram-units to do the calculation:

0.1064kg x 97.9m/s = 10.42 kgm/s

This appears to be a change in momentum for the entire system unless we consider that firing the arrow forward creates an equal and opposite force on the archer, himself. The sum of the new momentums needs to equal the sum of the original momentums, which was zero.

P_archer+ P_arrow = 0

P_archer+ 10.42kgm/s = 0

10.42kgm/s = -P_archer

P_archer = -10.42kgm/s

This results in an equal momentum for him, but going in the opposite direction, so he is going to slide in the opposite direction of the arrow.

P_archer = V_archer * M_archer = -10.42kgm.s

V * 105.9kg = -10.42kgm/s

V = -10.42kgm/s / 105.9kg

V = -0.09839 m/s or -98.39 mm/s

The negative velocity indicates that his direction of movement is opposite that of the arrow.