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?
To solve this problem, we need to use the principle of conservation of momentum. According to this principle, the total momentum of a system of objects remains constant if no external forces act on the system. In this case, we have two objects: the man and the canoe.
Here are the steps to solve the problem:
Step 1: Calculate the initial momentum of the system.
The initial momentum is the sum of the momenta of the man and the canoe when they are at rest.
Initial momentum = (mass of the man × velocity of the man) + (mass of the canoe × velocity of the canoe)
Since both the man and the canoe are initially at rest, their initial velocities are zero.
Initial momentum = (mass of the man × 0) + (mass of the canoe × 0) = 0
Step 2: Calculate the final momentum of the system.
The final momentum is the sum of the momenta of the man and the canoe after the man walks from the back to the front of the canoe.
Final momentum = (mass of the man × velocity of the man) + (mass of the canoe × velocity of the canoe)
Step 3: Apply the conservation of momentum principle.
According to the conservation of momentum principle, the initial momentum and the final momentum of the system should be equal.
Initial momentum = Final momentum
Since the initial momentum is zero, we can set the final momentum equal to zero.
Final momentum = 0
Therefore, we have:
(mass of the man × velocity of the man) + (mass of the canoe × velocity of the canoe) = 0
Step 4: Calculate the velocity of the man.
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.
The total distance the man walks is 0.75 m + 0.75 m = 1.5 m.
Since the time taken for the man to walk is the same as the time taken for the canoe to move, we can use the equation:
velocity = displacement / time
The displacement of the man is equal to the total distance he walks, which is 1.5 m.
We need to find the time taken for the man to walk the distance of 1.5 m.
To find the time taken, we need to know the velocity of the canoe. However, we don't have that information given in the problem. Therefore, we cannot determine the velocity of the man or the distance the canoe moves with the given information.
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant unless acted upon by an external force.
Let's break down the problem into two parts: before and after the man walks.
Before the man walks: The total momentum of the system consists of the momentum of the man and the momentum of the canoe. Since the system is initially at rest, the total momentum is zero.
After the man walks: The man exerts a force on the canoe, propelling it forward. The key to solving this problem is to determine the velocity of the man and the canoe after the man walks.
We can use the principle of conservation of momentum to calculate the velocity of the man and the canoe after the man walks.
According to the principle of conservation of momentum, the total momentum before the man walks is equal to the total momentum after the man walks.
Total momentum before = Total momentum after
(0 + 0) = (m1 * v1) + (m2 * v2)
where
m1 = mass of the man
v1 = velocity of the man after walking
m2 = mass of the canoe
v2 = velocity of the canoe after the man walks
The man and the canoe move in opposite directions, so their velocities have opposite signs.
m1 * v1 = -m2 * v2
Plugging in the given values:
m1 = 64.1 kg (mass of the man)
m2 = 61 kg (mass of the canoe)
64.1 kg * v1 = -61 kg * v2
Now, we need to find the ratio of the distances traveled by the man and the canoe:
The ratio of the distances traveled is the inverse of the ratio of their masses, according to the principle of conservation of momentum.
distance traveled by the canoe / distance traveled by the man = m1 / m2
We are given that the man walks 0.75 m from the back of the canoe to the front, so the distance traveled by the man is 0.75 m.
(distance traveled by the canoe) / 0.75 m = 61 kg / 64.1 kg
Now, solving for the distance traveled by the canoe:
distance traveled by the canoe = (61 kg / 64.1 kg) * 0.75 m
distance traveled by the canoe ≈ 0.711 m
Therefore, the canoe moves approximately 0.711 meters 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.