A 9.6 kg dog takes a nap in a canoe and wakes up to find the canoe has drifted out onto the lake but now is stationary. He walks along the length of the canoe at 0.53 m/s, relative to the water, and the canoe simultaneously moves in the opposite direction at 0.12 m/s. What is the mass of the canoe?
To find the mass of the canoe, we can use the principle of conservation of momentum. According to this principle, the total momentum before and after an event remains constant in the absence of external forces.
In this case, the dog and canoe are initially at rest, so the total momentum is zero. After the dog walks in the canoe, both the dog and canoe acquire momentum in opposite directions.
We can calculate the momentum separately for the dog and the canoe, and then apply the conservation of momentum to find the mass of the canoe.
The momentum of an object is given by the product of its mass and velocity.
Let's denote the mass of the dog as "m1," the mass of the canoe as "m2," the velocity of the dog as "v1," and the velocity of the canoe as "v2."
The dog's momentum, p1, is given by:
p1 = m1 * v1
The canoe's momentum, p2, is given by:
p2 = m2 * v2
According to the conservation of momentum principle, the total momentum after the event should be zero, so we have:
p1 + p2 = 0
Substituting the expressions for p1 and p2, we get:
m1 * v1 + m2 * v2 = 0
Now, let's plug in the given values:
m1 = 9.6 kg
v1 = 0.53 m/s
v2 = -0.12 m/s (negative because the canoe is moving in the opposite direction)
Substituting these values into the equation, we have:
9.6 kg * 0.53 m/s + m2 * (-0.12 m/s) = 0
Simplifying the equation gives us:
5.088 kg m/s - 0.12 m/s * m2 = 0
Now, isolate the term with m2:
-0.12 m/s * m2 = -5.088 kg m/s
Divide both sides of the equation by -0.12 m/s to solve for m2:
m2 = (-5.088 kg m/s) / (-0.12 m/s)
m2 ≈ 42.4 kg
Therefore, the mass of the canoe is approximately 42.4 kg.