A dog sits on the left end of a boat of length L that is initially adjacent to a dock to its right. The dog then runs toward the dock, but stops at the end of the boat. If the boat is H times heavier than the dog, how close does the dog get to the dock?
Ignore any drag force from the water.
To solve this problem, we can use the principle of conservation of momentum. Let's assume the mass of the dog is m, and the mass of the boat is Hm (since the boat is H times heavier than the dog).
Initially, the system (dog + boat) is at rest, and the momentum is zero. When the dog starts running towards the dock, it exerts a force on the boat in the opposite direction. According to Newton's third law, the boat exerts an equal and opposite force on the dog. However, since the mass of the boat is much larger than the mass of the dog, the effect on the boat's velocity is negligible.
Now, let's analyze the situation when the dog stops at the end of the boat. At this point, both the dog and the boat are moving towards the dock with the same velocity, v. The momentum of the system is now given by:
(mass of dog) * (velocity of dog) + (mass of boat) * (velocity of boat) = 0
m * v + Hm * (-v) = 0
Now, we can solve for the velocity:
v - Hv = 0
(1 - H) v = 0
Since v cannot be zero (otherwise the dog could not have moved), we have:
1 - H = 0
H = 1
This means that the dog gets as close to the dock as the length of the boat. In other words, the dog stops at the end of the boat, adjacent to the dock.
To determine how close the dog gets to the dock, we need to consider the conservation of momentum.
The initial momentum of the system, consisting of the dog and the boat, is zero since there is no initial motion. When the dog runs toward the dock and stops at the end of the boat, the momentum of the system is still zero.
Let's assume the initial velocity of the dog is v and the final velocity of the boat is V. Since the dog stops at the end of the boat, its final velocity is zero.
By applying the conservation of momentum, we have:
(mass of the dog) * v = (mass of the boat) * V
Since the boat is H times heavier than the dog, we can express the mass of the boat as H * (mass of the dog).
Therefore, the equation becomes:
(mass of the dog) * v = H * (mass of the dog) * V
(mass of the dog) cancels out from both sides of the equation, resulting in:
v = H * V
Now, we can solve for V:
V = v / H
Since the boat travels to the right and the dog stops at the end of the boat, the velocity of the boat is positive. Therefore, V > 0.
To determine how close the dog gets to the dock, we need to find the distance traveled by the boat. The distance traveled by the boat is equal to the length of the boat, L. Therefore, the dog gets L units close to the dock.
In conclusion, the dog gets L units close to the dock.