A canoe of mass 38 kg lies at rest in still water. A man and a woman are at opposite ends of the canoe 4.0 m apart and symmetrically located with respect to the canoe’s centre (which is also its centre of mass). The mass of the man is 65 kg and the woman’s mass is smaller.
The two people then change places and the man observes that the canoe shifts a distance 0.20 m relative to the water. What is the woman’s mass
68.7
To find the woman's mass, we can use the principle of conservation of momentum.
The initial momentum of the system (person + canoe) is zero since they are at rest. When the man and woman switch places, the canoe shifts a distance relative to the water. This shift indicates a change in momentum in the horizontal direction.
The momentum before the switch is given by:
Initial momentum = mass of the man * velocity of the man + mass of the woman * velocity of the woman
Since the canoe is initially at rest, the velocity of both the man and woman is zero. Therefore, the initial momentum is simply equal to the mass of the woman * 0.
After the switch, the momentum is given by:
Final momentum = mass of the man * velocity of the woman + mass of the woman * velocity of the man
From the problem statement, we know that the canoe shifts a distance of 0.20 m relative to the water. This means that the velocity of the woman is 0.20 m / time taken to switch places.
Since the man and woman are symmetrically located with respect to the canoe's center of mass, the velocity of the man is equal in magnitude but opposite in direction to the velocity of the woman. Therefore, the velocity of the man is -0.20 m / time taken to switch places.
Substituting the values into the final momentum equation, we get:
Final momentum = 65 kg * (-0.20 m / time taken to switch places) + mass of the woman * (0.20 m / time taken to switch places)
Now, since the initial momentum is zero (since the canoe is at rest initially) and momentum is conserved, the final momentum must also be zero.
Therefore, we can set the final momentum equation equal to zero and solve for the mass of the woman:
0 = -0.20 * 65 + mass of the woman * 0.20
Simplifying the equation, we get:
0 = -13 + 0.20 * mass of the woman
Rearranging the equation, we can isolate the mass of the woman:
0.20 * mass of the woman = 13
Dividing both sides by 0.20, we get:
mass of the woman = 13 kg
Therefore, the mass of the woman is 13 kg.