Two people, one of mass 85 kg and the other of mass 50 kg, sit in a rowboat of mass 90 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.5 m apart from each other, now exchange seats. How far does the boat move?
Here is the key. Center of gravity stays put. So two folks move, and between them, they move the cg, but the boat moves back that far. (Equal and opposite reaction)
so apparently 35 kg (85-50) is shifted 3.5m.
cg shift? 35/(90+85+50)*3.5 m
figure that out. Then, the boats shifts back that far.
To solve this problem, we need to use the principle of conservation of momentum. The momentum before the exchange of seats is equal to the momentum after the exchange.
Initially, the boat is at rest, so the total momentum is zero.
The momentum of each person can be calculated using the formula: momentum = mass × velocity.
Let's assume the person with a mass of 85 kg moves to the other end of the boat. To calculate their final velocity, we can use the equation: momentum = mass × velocity. Since the boat is at rest initially, and there is no external force acting on it, the change in momentum of the person will be equal in magnitude to the change in momentum of the boat.
The person's initial momentum = mass × velocity = 85 kg × 0 m/s = 0 kg•m/s (since the person is initially at rest).
The person's final momentum = mass × final velocity.
To find the final velocity, we can rearrange the equation for momentum:
final velocity = final momentum / mass.
The final momentum for the person is 0 kg•m/s since they started and ended at rest.
By substituting the known values into the equation, we find:
final velocity = 0 kg•m/s / 85 kg = 0 m/s.
Therefore, the person's final velocity is 0 m/s, indicating that they didn't move.
Since the person with a mass of 50 kg exchanged seats, the same principle applies. The final momentum for that person is also 0 kg•m/s, and their final velocity is 0 m/s.
So, we can conclude that the final momentum of the boat is 0 kg•m/s as well.
Now, let's determine the initial momentum of the boat. It is the combined momentum of the two people sitting at opposite ends.
Using the formula momentum = mass × velocity:
Initial momentum of the 85 kg person = 85 kg × 0 m/s = 0 kg•m/s.
Initial momentum of the 50 kg person = 50 kg × 0 m/s = 0 kg•m/s.
So, the initial momentum of the boat is 0 kg•m/s as well.
According to the principle of conservation of momentum (initial momentum = final momentum), the boat, with a mass of 90 kg, will have no net change in momentum before and after the seat exchange. Therefore, the boat will remain at rest, and it will not move.
To find out how far the boat moves, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system (in this case, the rowboat and the people) remains constant if no external forces act on it.
Initially, let's consider the momentum of the system:
Initial momentum = (mass of person 1 * velocity of person 1) + (mass of person 2 * velocity of person 2) + (mass of boat * velocity of the boat) = 0
Since the boat is initially at rest, its velocity is 0.
Now, when the people exchange seats, they effectively change their positions in the boat. Assuming no external forces act on the system, the total momentum of the system should remain constant.
Let's denote the final velocity of person 1 as v1, the final velocity of person 2 as v2, and the final velocity of the boat as vb. Since the boat moves in the opposite direction to the motion of the people to conserve momentum, we can write:
(mass of person 1 * v1) + (mass of person 2 * v2) + (mass of boat * vb) = 0
We know the masses of the people and the boat, so we can substitute those values:
(85 kg * v1) + (50 kg * v2) + (90 kg * vb) = 0
Now, let's consider the distances traveled by the people when they exchange seats. Person 1 moves a distance of 3.5 m to reach person 2's initial position, and person 2 moves a distance of 3.5 m to reach person 1's initial position.
By the principle of conservation of momentum, we can say that:
Mass of person 1 * distance traveled by person 1 = Mass of person 2 * distance traveled by person 2
(85 kg * 3.5 m) = (50 kg * d2)
Simplifying, we get:
d2 = (85 kg * 3.5 m) / 50 kg
d2 = 5.95 m (rounded to two decimal places)
So, person 2 travels a distance of approximately 5.95 meters.
Since the boat moves in the opposite direction to the motion of the people, the boat also moves a distance of 5.95 meters.
Therefore, the boat moves approximately 5.95 meters when the people exchange seats.