A dog, with a mass of 8.0 kg, is standing on a flatboat so that he is 26.8 m from the shore. He walks 8.6 m on the boat toward the shore and then stops. The boat has a mass of 36.0 kg. Assuming there is no friction between the boat and the water, how far is the dog from the shore now?
Use the fact that the center of mass of boat + dog stays in the same place.
To determine how far the dog is from the shore after walking on the boat, we can use the principle of conservation of momentum.
First, let's consider the initial momentum of the system, which includes both the dog and the boat. The momentum is given by the product of mass and velocity. Since the dog is initially at rest, its initial momentum is zero (mass of 8.0 kg multiplied by velocity of 0 m/s is zero). The initial momentum of the boat is also zero as it is not moving initially (mass of 36.0 kg multiplied by velocity of 0 m/s is zero).
Next, let's consider the momentum after the dog walks on the boat. Since there is no external force acting on the system and no friction between the boat and the water, the momentum of the system should remain conserved.
Let's denote the new velocity of the dog as v_d and the new velocity of the boat as v_b. The total momentum of the system after the dog walks on the boat is given by:
(mass of dog * velocity of dog) + (mass of boat * velocity of boat) = 0
Since the dog walks towards the shore, its new velocity is positive. On the other hand, since the boat is pushed in the opposite direction to the dog's motion, its new velocity is negative.
Let's use this information to set up an equation:
(8.0 kg * v_d) + (36.0 kg * (-v_b)) = 0
Given that the initial distance between the dog and the shore is 26.8 m, and the dog walks 8.6 m towards the shore, the final distance between the dog and the shore can be obtained using the equation:
final distance = initial distance + distance walked on the boat
final distance = 26.8 m + 8.6 m
Simplifying, we can combine these equations to solve for the unknown, final distance:
(8.0 kg * v_d) + (36.0 kg * (-v_b)) = 0
(8.0 kg * v_d) = (36.0 kg * v_b)
v_d = (36.0 kg * v_b) / 8.0 kg
Substituting the known values, we can find v_d:
v_d = (36.0 kg * 8.6 m/s) / 8.0 kg
v_d = 38.7 m/s
Finally, the final distance from the shore can be calculated as follows:
final distance = initial distance + distance walked on the boat
final distance = 26.8 m + 8.6 m
final distance = 35.4 m
Therefore, the dog is 35.4 meters from the shore after walking on the boat.