Your fhe brother (mass 144.1 kg) is canoeing on utah lake with his sister (mass 81.6 kg). he had the only paddle but managed to drop it in the lake. it has drifted just out of his reach along the long axis of the canoe. his sister has taken physics 121 and suggests that if they trade places she might be able to reach the paddle. due to the high "slime" content of the lake and the slick surface of the canoe, the canoe moves in the water without any friction or drag. the passengers sit 2.03 meters apart and the canoe has a mass of 48.7 kg. after the passengers switch places, how much closer to the paddle (which does not move) is the end of the canoe? (enter a positive number.)

Explanation:

This problem can be solved using the center of mass of the system.

As there are no external forces in the horizontal direction, and the external forces in the vertical direction, the gravitational force and the buoyancy,  cancel each other, we know:

where M is the total mass of the system, and is the linear momentum of the system. This implies:

so

the linear momentum of the system is:

where is the position of the center of mass. As the mass in this problem is constant, this implies that the position of the center of mass is conserved.

For a system formed by i particles, each of mass and position , the center of mass can be found with the equation:

where M is the total mass of the system, found with the equation:

For this problem, the total mass of the system is :

To find the positions, we need an frame of reference.  Putting the origin of coordinate at the position of the sister.

The brother is 2.03 meters apart in the positive direction:

and the center of mass of the canoe must be at half this distance:

taking all this, the center of mass of the system is at :

After the switch, we found that the center of mass has to be displaced. If its displaced a distance d, we find that the new positions must be:

where the brother and the sister had switched the original position. Now, the center of mass will be located at:

And the equation is :

And this is how much closer the paddle is to the end of the canoe.