A Walk on the Pond

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 1.0 m/s and the girl has a mass of 20 kg and was initially running at 2.0 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?

Solution

System: Boy and girl as point particles.

Interactions: Impulse from external influences is neglected during the collision under the assumption that the collision is instantaneous. After the collision, the system experiences external influences from the earth (gravity, conservative) and the ice (normal force and friction, each non-conservative).

Models: Momentum and External Force followed by [Mechanical Energy and Non-Conservative Work].

Approach:

Diagrammatic Representation, Part 1

We begin by analyzing the collision using momentum. When a problem gives a relative angle, it is important to develop a coordinate system to orient ourselves as we solve. We therefore 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.

Mathematical Representation, Part 1

With our picture developed, we can write the equations of constant momentum, since we are assuming that the collision occurs over such a short time that external forces create a negligible impulse.
{latex}\begin{large}[ p^{B}_{x,i} + p^{G}_{x,i} = p^{B+G}_{x,f}]
[p^{B}_{y,i} + p^{G}_{y,i} = p^{B+G}_{y,f} ]\end{large}{latex}

Rewriting in terms of the masses and velocities, and substituting the appropriate zeros gives:
{latex}\begin{large}[ m^{B}v^{B}_{x,i} + m^{G}v^{G}_{x,i} = (m^{B}+m^{G})v_{x,f} ]
[ m^{G}v^{G}_{y,i} = (m^{B}+m^{G})v_{y,f} ]\end{large}{latex}

We have all the givens we need to solve these equations directly for the final velocity of the system:
{latex}\begin{large}[ v_{x,f} = \frac{m^{B}v^{B}_{x,i}+m^{G}v^{G}_{x,i}}{m^{B}+m^{G}} = \mbox{1.24 m/s}]
[ v_{y,f} = \frac{m^{G}v^{G}_{y,i}}{m^{B}+m^{G}} = \mbox{0.808 m/s} ]\end{large}{latex}

so that the magnitude of the final velocity is:
{latex}\begin{large}[ v_{f} = \sqrt{(\mbox{1.24 m/s})^{2+(\mbox{0.808 m/s})}{2}} = \mbox{1.48 m/s} ]\end{large}{latex}



Diagrammatic Representation, Part 2

This final velocity for the collision of the boy and girl is the initial velocity for their post-collision slide. We can now consider a free body diagram for the boy+girl system as they slide across the ice. Assuming the pond is flat, the [gravitational potential energy] will be constant and the normal force will do no work. Friction will do negative work, as it is directed at 180° from the direction of motion of the system. Since only one force can do work, the question of how far the system slides after the collision is easily addressed using work and energy.

Mathematical Representation, Part 2

The relevant equation is:

Considering the free body diagram plus the fact that the boy and girl are not accelerating in the z-directtion, it is clear that the normal force and the weight of the system must balance. Further, we want to know how far the boy and girl slide before they stop, indicating that v_f should be zero for this part of the problem. Thus, we can write:

Again, recognizing that the initial velocity for the slide is the final velocity of the collision, we can solve to find: