A man with a mass(mm) of 71.9 kg stands up in a 95-kg canoe(mc) of length(l) 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?
No external forces on system so center of mass continues at constant speed of zero so does not move
how far did he walk in the canoe?
4.00 - 1.5 = 2.5 m
distance he moved * his mass = distance canoe moved * its mass
2.5 *71.9 = 95 x
x = 2.5 * 71.9 / 95
To solve the problem, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the man walks must be equal to the total momentum after the man walks.
Now, let's break down the problem into steps:
Step 1: Calculate the initial momentum.
The initial momentum is the combined momentum of the man and the canoe before the man walks. The formula for momentum is:
momentum = mass × velocity
Since the canoe is floating on the water, its initial velocity is zero. So the momentum of the canoe (mc) is:
momentum(mc) = mc × 0 = 0
The man's momentum (mm) can be calculated by multiplying his mass (mm) by his velocity (vmm). We need to find the velocity (vmm) first, using the concept of center of mass.
Step 2: Calculate the velocity of the man's center of mass.
The concept of center of mass states that when a person walks from one point to another, their center of mass moves by the same distance. In this case, the man 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, covering a total distance of 1.5 m.
Since we know the distance traveled and the time it took (we assume the time it took is negligible), we can calculate the velocity of the man's center of mass using the formula:
velocity = distance / time
In this case, the velocity of the man's center of mass (vmm) is:
vmm = 1.5 m / 0 seconds = 0 m/s
Step 3: Calculate the final momentum.
The final momentum is the combined momentum of the man and the canoe after the man walks. The man's momentum (mm) is calculated by multiplying his mass (mm) by his velocity (vmm), which we calculated as zero.
momentum(mm) = mm × vmm = mm × 0 = 0
The canoe's momentum (mc) has not changed, so it is still zero.
Step 4: Apply the law of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the man walks must be equal to the total momentum after the man walks.
initial momentum (mc + mm) = final momentum (mc + mm)
0 + 0 = 0 + 0
Since the initial and final momenta are both zero, we can conclude that the canoe does not move when the man walks along it. Hence, the distance the canoe moves is zero.