What distance does a canoe move when a woman walks across it?

[anthonych414]

Homework Statement

A 45.0-kg woman stads up in a 60-kg canoe 5.00 m long. She walks from a point 1.00 m from one end to a point 1.00 m from the other end. If you ignore resistance to motion of the canoe in water, how far does the canoe move during this process.

Homework Equations

p=mv
J=p2-p1
p1=p2

The Attempt at a Solution

I guess I should try to make the momentum before and after equal, but I don't know how to relate that to the distance.

 

[Infinitum]

Since there is no external force acting on the system in the direction of motion, you can use the fact that the position of center of mass remains unchanged. Draw out a diagram, and frame equations making sure that the COM stays same.

 

[tiny-tim]

hi anthonych414!

hint: what happens to the centre of mass?

 

[anthonych414]

Right, the center of mass! I can't believe I didn't think of it! I feel silly... Anyway thanks for the help guys :D

 

[Nickie]

I would approach this problem by first identifying the relevant equations and principles that apply to this situation. In this case, the principle of conservation of momentum and the equation p=mv are applicable.

To solve this problem, we need to determine the initial and final momenta of the system. The initial momentum (p1) can be calculated by multiplying the mass of the woman (m1 = 45 kg) by her initial velocity (v1 = 0 m/s), which is zero since she is initially standing still. The initial momentum of the canoe (p2) can be calculated by multiplying its mass (m2 = 60 kg) by its initial velocity (v2 = 0 m/s). Therefore, p1 = 0 and p2 = 0.

Next, we need to determine the final momentum of the system after the woman has walked across the canoe. The final momentum (p2) can be calculated by multiplying the total mass of the system (m1 + m2 = 105 kg) by the final velocity (v2). Since the woman has walked from one end to the other, the final velocity of the system is the same as the velocity of the woman, which we can calculate using the equation v = d/t, where d is the distance she walked (5.00 m) and t is the time it took her to walk that distance.

However, since we are ignoring resistance to motion of the canoe in water, we can assume that the time it took for the woman to walk across the canoe is very short and can be considered negligible. Therefore, the final velocity of the system (and the woman) is very large, but since we are only interested in the change in momentum, we can use the equation p2 = m2v2, where m2 is the mass of the canoe and v2 is the final velocity of the canoe.

Now, to solve for the final velocity of the canoe, we can use the principle of conservation of momentum, which states that the initial momentum of a system is equal to the final momentum of the system. In this case, we know that p1 = p2, so we can set up the equation m1v1 = m2v2 and solve for v2. Plugging in the known values, we get v2 = (m1v1)/m2 = (45 kg)(0 m/s)/(60 kg) =