A man with a mass of 64.1 kg stands up in a 61-kg canoe of length 4.00 m floating on water. He walks from a point 0.75 m from the back of the canoe to a point 0.75 m from the front of the canoe. Assume negligible friction between the canoe and the water. How far does the canoe move?
The canoe will move so that the center of mass of man plus canoe stays in the same place.
The center of mass in the beginning is a distance X from center, in the back portion.
64.1*1.25 + 61*0 = 125.1 X
X = 1.13 m
When the man walks to within 0.75 m of the front of the canoe, he will have moved 2.5 m relative to the canoe. The center of mass will then be 1.13 m to the front of center.
For the center of mass to remain in the same place relative to the water, the cane must move 2.26 m to the rear
To find out how far the canoe moves when the man walks from one end to the other, we can use the concept of conservation of momentum.
First, let's find the initial momentum of the system before the man walks. The initial momentum is the sum of the momentum of the man and the momentum of the canoe.
The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v): p = m * v.
In this case, the man is initially at rest, so his momentum is zero. The momentum of the canoe is given by: p(c) = m(c) * v(c), where m(c) is the mass of the canoe and v(c) is the velocity of the canoe.
Since there is no external force acting on the system, the total momentum before and after the man's movement must be the same.
Now, let's find the final momentum of the system after the man walks. When the man moves from one end to the other, the canoe moves in the opposite direction.
Let's assume the velocity of the canoe after the man walks is v(final). The man's final velocity is v(final) as well, but in the opposite direction.
So, the final momentum of the system is: p(final) = (m + m(c)) * v(final) - m * v(final).
Since the total momentum is conserved, we can set the initial momentum equal to the final momentum:
0 = (m + m(c)) * v(final) - m * v(final).
Now we can solve for v(final):
0 = (m + m(c) - m) * v(final).
m cancels out:
0 = m(c) * v(final).
Since we assume the final velocity of the canoe is zero (since it stops moving), we have:
0 = m(c) * 0.
This means the final velocity of the canoe is zero, and therefore it does not move when the man walks from one end to the other.
Thus, the canoe does not move at all when the man walks from a point 0.75 m from the back to a point 0.75 m from the front of the canoe.