A man (mass 120.5 kg) is canoeing with his friend (mass 81.6 kg). He had the only paddle but dropped it in the lake. It has drifted just out of his reach along the long axis of the canoe. His friend suggests that if they trade places he might be able to reach the paddle. The canoe moves in the water without any friction or drag. The passengers sit 2.03 meters apart and the canoe has a mass of 48.7 kg. After the passengers switch places, how much closer to the paddle (which does not move) is the end of the canoe?
I'm not really sure where to start or how to go about this problem. Any steps to start out would be appreciated. Thank you :)
To solve this problem, we can start by applying the principle of conservation of momentum. The total momentum before the passengers switch places should be equal to the total momentum after they switch places. Let's break down the problem into steps:
Step 1: Calculate the initial momentum before the passengers switch places.
- The momentum of the man before switching places can be calculated using the formula: momentum = mass * velocity.
- Since the man is initially at rest, his momentum is zero.
- The momentum of the friend before switching places can also be calculated using the same formula: momentum = mass * velocity.
- Since the friend is initially at rest as well, his momentum is also zero.
- So, the total initial momentum before switching places is zero.
Step 2: Calculate the final momentum after the passengers switch places.
- After switching places, the man and the friend will have new velocities.
- However, the total momentum of the system should remain zero, as there is no external force acting on it.
- Therefore, the final momentum should also be zero.
Step 3: Use the principle of conservation of momentum to calculate the new velocities.
- Let's assume the velocity of the man after switching places is "Vm" and the velocity of the friend after switching places is "Vf."
- Since the canoe is at rest initially, the total mass of the system (man + friend + canoe) is conserved.
- We can write the conservation of momentum equation as: (mass of man * Vm) + (mass of friend * Vf) = 0.
Step 4: Solve for the new velocities.
- Substituting the given masses, we have: (120.5 kg * Vm) + (81.6 kg * Vf) = 0.
- Rearranging the equation to solve for Vm, we get: Vm = -(81.6 kg * Vf) / 120.5 kg.
Step 5: Calculate the distance the canoe moves after switching places.
- Since the passengers switch places, the total mass remains the same.
- The new center of mass will be the same as the initial one.
- Therefore, the distance of the end of the canoe from the paddle will remain unchanged.
Step 6: Calculate how much closer the end of the canoe is to the paddle.
- Since the distance remains the same after switching places, the end of the canoe will be the same distance from the paddle as before the switch.
In conclusion, after the passengers switch places, the distance from the end of the canoe to the paddle remains unchanged.