A man with a mass of 58 kg stands up in a 69-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.)
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To find how far the canoe moves, we can use the principle of conservation of momentum. The initial momentum of the man and the canoe should be equal to their final momentum.
Let's label the point where the man starts as A (0.75 m from the back of the canoe) and the point where the man ends as B (0.75 m from the front of the canoe).
1. Calculate the initial momentum:
The initial momentum, p_initial, is the sum of the momentum of the man (p_man) and the momentum of the canoe (p_canoe).
p_initial = p_man + p_canoe
The momentum can be calculated as the product of the mass (m) and velocity (v):
p = m * v
For the man at point A, the velocity will be zero since he is stationary relative to the canoe.
So, p_man_initial = 0 kg*m/s.
For the canoe, the velocity is determined by the movement of the man.
The mass of the canoe is given as 69 kg (m_canoe), and its velocity (v_canoe_initial) is unknown.
Therefore, p_canoe_initial = m_canoe * v_canoe_initial.
2. Calculate the final momentum:
The final momentum, p_final, will be the sum of the momentum of the man (p_man_final) and the momentum of the canoe (p_canoe_final).
p_final = p_man_final + p_canoe_final
Since the man walks from A to B, he acquires a forward velocity.
The mass of the man is given as 58 kg (m_man), and his velocity (v_man_final) can be calculated using the formula: v_final = (distance)/(time). We need to determine the time taken by the man to walk from A to B.
The distance between A and B is 0.75 m.
The time taken by the man can be calculated using the formula: time = (distance)/(velocity).
Since the man starts from rest, his initial velocity (v_man_initial) is zero.
Therefore, time = (distance)/(v_man_final - v_man_initial) = 0.75 m / v_man_final.
Now we need to determine the velocity of the man at point B.
We can use the concept of conservation of energy since the man is not losing or gaining any external energy.
The potential energy lifted by the man walking from A to B should be equal to the kinetic energy gained.
The potential energy lifted can be calculated using the formula: potential energy = m * g * h, where h is the height.
In this case, the height (h) of the man walking can be approximated as the distance between A and B, which is 0.75 m.
The gravitational acceleration (g) is approximately 9.8 m/s^2.
The kinetic energy gained can be calculated as: kinetic energy = 0.5 * m * (v_man_final)^2.
Setting the potential energy equal to the kinetic energy:
m * g * h = 0.5 * m * (v_man_final)^2
Simplifying, we find:
g * h = 0.5 * (v_man_final)^2
Solving for v_man_final:
v_man_final = sqrt(2 * g * h)
Substituting the given values:
v_man_final = sqrt(2 * 9.8 m/s^2 * 0.75 m) = 3.83 m/s.
Now we have the velocity of the man at point B.
Therefore, p_man_final = m_man * v_man_final.
For the canoe, the momentum remains unchanged since there are no external forces acting on it.
So, p_canoe_final = p_canoe_initial.
3. Equate the initial and final momentum:
p_initial = p_final
0 kg*m/s + m_canoe * v_canoe_initial = m_man * v_man_final + p_canoe_initial
Since p_canoe_final = p_canoe_initial, we can rewrite it as:
m_canoe * v_canoe_initial = m_man * v_man_final + m_canoe * v_canoe_initial
Canceling out m_canoe * v_canoe_initial from both sides:
0 = m_man * v_man_final
Solving for the velocity of the man:
v_man_final = 0 m/s.
This implies that the man didn't move in the horizontal direction.
Therefore, the canoe will also not move horizontally.
So, the distance the canoe moves is 0 meters.