A man with a mass of 50 kg stands up in a 66-kg canoe of length 4.0 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? (Assume the canoe has a uniform density such that its center of mass location is at the center of the canoe.)
Use the fact that the center mass of the man and canoe (together) does not move, as seen from shore. This is a result of total momentum conservation.
Initially, the CM is 1.25*50/116 = 0.514 m back of the center of the canoe. After moving, it is 0.514 m from forward of center.
For the CM to stay in the same place (as seen from shore), the canoe must move 2 x 0.514 = 1.028 m backwards.
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To solve this problem, we need to use the principle of conservation of momentum. The total momentum before the man starts walking is equal to the total momentum after he walks.
Let's break down the steps to solve the problem:
Step 1: Find the initial momentum
The initial momentum of the system is given by the product of the mass and velocity of the canoe before the man starts walking. Since the canoe is initially at rest, its initial momentum is zero.
Initial momentum = 0 kg * m/s
Step 2: Find the final momentum
To find the final momentum, we need to consider the momentum of both the man and the canoe when the man reaches his final position.
The mass of the canoe is 66 kg, the mass of the man is 50 kg, and the length of the canoe is 4.0 m. The initial position of the man is 0.75 m from the back of the canoe, and the final position of the man is 0.75 m from the front of the canoe.
The final momentum of the system is given by the product of the total mass and the velocity of the system. Since the total mass does not change, the final momentum of the system will be equal to the initial momentum of the system.
Final momentum = 0 kg * m/s
Step 3: Find the distance the canoe moves
Since the final momentum is equal to the initial momentum, we know that the momentum of the man and the canoe is canceled out. This means that the momentum gained by the man walking forward is exactly balanced by the momentum lost by the canoe moving backward.
Since the mass of the canoe is greater than the mass of the man, the distance the canoe moves will be less than the distance the man moves.
Therefore, the distance the canoe moves is equal to the difference between the initial and final positions of the man, which is:
Distance = (0.75 m + 0.75 m) - (0.75 m - 0.75 m)
= 1.5 m
So, the canoe moves a distance of 1.5 meters.
To determine how far the canoe moves when the man walks, we need to consider the principle of conservation of momentum.
First, let's find the initial momentum of the man and the canoe system. Momentum is given by the product of an object's mass and its velocity.
The initial total momentum (before the man walks) can be calculated by:
Initial Total Momentum = (Mass of the man * Velocity of the man) + (Mass of the canoe * Velocity of the canoe)
Since the man is initially at rest, the initial velocity of the man is zero. However, the velocity of the canoe is unknown. We can solve for it using the principle of the conservation of momentum.
According to the principle of the conservation of momentum, the total momentum before and after any interaction remains constant, provided no external forces act on the system.
When the man walks from the back to the front of the canoe, the system consisting of the man and the canoe is isolated, meaning no external forces act on it. Therefore, the total momentum of the system will remain constant.
Let's denote the velocity of the canoe as V(canoe) and the velocity of the man after he walks as V(man).
According to the principle of conservation of momentum, the initial total momentum is equal to the final total momentum:
0 = (Mass of the man * V(man)) + (Mass of the canoe * V(canoe))
Given that the mass of the man (m(man)) is 50 kg, and the mass of the canoe (m(canoe)) is 66 kg, we rewrite the equation as:
0 = (50 kg * V(man)) + (66 kg * V(canoe))
Now, let's find the final velocity of the man, V(man), using the concept of center of mass. Since the canoe has a uniform density and its center of mass is at the center of the canoe, we can conclude that the center of mass of the man and the canoe system doesn't move when the man walks.
In other words, the initial center of mass of the system coincides with the final center of mass after the man walks.
To find the final velocity of the man, we use the equation:
m(man) * (0.75 m) = (m(man) + m(canoe)) * (0.5 m)
Substituting the values, we have:
50 kg * (0.75 m) = (50 kg + 66 kg) * (0.5 m)
37.5 kg m = 116 kg m
Now, we can solve for V(man):
V(man) = 37.5 kg m / 116 kg ≈ 0.323 m/s
Substituting this value into the conservation of momentum equation, we can solve for V(canoe):
0 = (50 kg * 0.323 m/s) + (66 kg * V(canoe))
- 16.15 kg m/s = 66 kg * V(canoe)
V(canoe) = - 16.15 kg m/s / 66 kg ≈ -0.245 m/s
The negative sign indicates that the canoe moves in the opposite direction. To find the distance the canoe moves, we multiply the velocity of the canoe by the time it takes for the man to walk from the back to the front of the canoe.
Given that the length of the canoe is 4.0 m and the man takes 0.75 m to walk from the back to the front, the distance the canoe moves is:
Distance = (velocity of the canoe) * (time taken)
Distance = -0.245 m/s * (0.75 m / -0.245 m/s) ≈ -0.75 m
Since distance cannot be negative, we can take the absolute value of the result, giving us the final distance:
Distance = |-0.75 m| ≈ 0.75 m
Therefore, the canoe moves approximately 0.75 meters.