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. The total momentum before the dog starts walking should be equal to the total momentum after the dog starts walking.
Before the dog starts walking, the canoe and the dog have a combined momentum of zero since they are stationary.
After the dog starts walking, the momentum of the dog and the canoe should still be zero, but now they have opposite velocities.
Let's break down the problem step by step:
1. Determine the momentum of the dog before and after the walk:
- Before the dog starts walking, the momentum of the dog is (0 kg⋅m/s) since it's stationary.
- After the dog starts walking, the momentum of the dog is given by the formula: momentum = mass × velocity. Assuming the dog's mass is 9.6 kg and its velocity relative to the water is 0.53 m/s, we can calculate the momentum of the dog using the formula: momentum = 9.6 kg × 0.53 m/s.
2. Determine the momentum of the canoe before and after the walk:
- Before the dog starts walking, the momentum of the canoe is (0 kg⋅m/s) since it's stationary.
- After the dog starts walking, the canoe's velocity is in the opposite direction with a magnitude of 0.12 m/s. Therefore, the momentum of the canoe is given by the formula: momentum = mass × velocity. Let's assume the mass of the canoe is M kg, then its momentum is M kg × (-0.12 m/s) since its velocity is in the opposite direction.
3. Apply the principle of conservation of momentum:
According to the conservation of momentum, the total momentum before and after the dog starts walking must be the same. Therefore, we can set up an equation:
momentum of the dog before = momentum of the dog after + momentum of the canoe after.
0 kg⋅m/s = (9.6 kg × 0.53 m/s) + (M kg × (-0.12 m/s)).
4. Solve for the mass of the canoe (M):
Rearrange the equation and solve for M:
M kg × (-0.12 m/s) = -(9.6 kg × 0.53 m/s).
M kg = (9.6 kg × 0.53 m/s) / 0.12 m/s.
Calculate the value of M (the mass of the canoe) using the given numbers:
M kg = (9.6 kg × 0.53 m/s) / 0.12 m/s.
By evaluating this expression, you can determine the mass of the canoe. After performing the calculation, the mass of the canoe should be approximately 42 kg.