While a family is taking a walk on a frozen pond, the two small children (a boy and a girl) manage to run into each other. They become entangled, resulting in a totally inelastic collision. The boy has a mass of 15 kg and was initially running at 3.0 m/s and the girl has a mass of 20 kg and was initially running at 5.0 m/s. Before their collision, the relative angle between their velocities was 45°. Assuming that the coefficient of friction between the resulting boy+girl combination and the ice is 0.15, how far do they slide after the collision before coming to rest?
System: Boy and girl as point particles. External influences are neglected during the collision under the assumption that collision forces dominate. After the collision, the system experiences external influences from the earth (gravity) and the ice (normal force and friction).
Models: [Momentum and Impulse] followed by the [Work-Energy Theorem].
Approach: When a problem gives a relative angle, it is important to develop a coordinate system to orient ourselves as we solve. Thus, we begin with a picture. We have arbitrarily assigned the boy to move along the x-axis, and the girl to have positive x- and y-velocity components.
With our picture developed, we can write the equations of constant momentum, since we are assuming that during the collision external forces are negligible compared to the collision forces.
\begin
[ p^
_
+ p^
_
= p^
_
]
[p^
_
+ p^
_
= p^
_
]\end
Rewriting in terms of the masses and velocities, and substituting the appropriate zeros gives:
\begin
[ p^{