A boy of mass 65.3 kg is about to disembark from a canoe of mass 36.1 kg. The canoe is initially at rest, with the bow just touching the dock (see figure). The center of the canoe is 3.0 m behind the bow. As the boy moves forward 6.0 m to the bow, the canoe moves away from the dock.
(a) How far is the boy from the dock when he reaches the bow?
To determine the distance of the boy from the dock when he reaches the bow, we can use the law of conservation of momentum. The total momentum before and after the boy moves to the bow should remain the same.
Let's denote the final velocity of the boy and the canoe as Vf, the initial velocity of the canoe as Vc, and the final velocity of the canoe as Vcf. We can calculate these velocities using the equation:
(mass of boy × final velocity of boy) + (mass of canoe × final velocity of canoe) = (mass of boy × initial velocity of boy) + (mass of canoe × initial velocity of canoe)
The mass of the boy is given as 65.3 kg, and the mass of the canoe is given as 36.1 kg. The initial velocity of the canoe is 0 m/s since it is initially at rest.
Plugging these values into the equation:
(65.3 kg × Vf) + (36.1 kg × Vcf) = (65.3 kg × 0 m/s) + (36.1 kg × 0 m/s)
Simplifying, we get:
65.3 kg × Vf + 36.1 kg × Vcf = 0 kg·m/s
Now, let's calculate the velocity of the canoe when the boy reaches the bow. The boy moves forward 6.0 m, so the displacement of the canoe with respect to the dock is also 6.0 m. We can use the equation:
Vcf = Vc - (displacement of canoe / time)
The initial position of the canoe is the center, which is 3.0 m behind the bow. Therefore, the initial position of the canoe with respect to the dock is -3.0 m.
Plugging in the values:
Vcf = 0 m/s - (6.0 m / time)
Now, substituting Vcf in the momentum equation:
65.3 kg × Vf + 36.1 kg × (0 m/s - 6.0 m / time) = 0 kg·m/s
We now have two variables, Vf and time. We need another equation to solve for both variables.
Since the boy moves forward 6.0 m to the bow, his displacement with respect to the dock is also 6.0 m.
Now, we can use the equation for average velocity:
Average Velocity = Total Displacement / Total Time
The total displacement is 6.0 m, and we can consider the total time to be the time taken for the boy to reach the bow.
Rearranging the equation:
Total Time = Total Displacement / Average Velocity
In this case, the average velocity is the velocity of the boy, which is given by:
Velocity of boy = displacement of boy / time
The displacement of the boy is 6.0 m, as mentioned earlier. Substituting this into the equation:
Velocity of boy = 6.0 m / time
Now, we have an expression for the velocity of the boy in terms of time.
Substituting this velocity into the momentum equation:
65.3 kg × (6.0 m / time) + 36.1 kg × (0 m/s - 6.0 m / time) = 0 kg·m/s
Simplifying and solving for time:
(65.3 kg × 6.0 m) / time - (36.1 kg × 6.0 m) / time = 0
(65.3 kg × 6.0 m) - (36.1 kg × 6.0 m) = 0
(65.3 kg - 36.1 kg) × 6.0 m = 0
29.2 kg × 6.0 m = 0
175.2 kg·m = 0
Since the equation leads to 0 = 0, it implies that time can have any value.
Therefore, when the boy reaches the bow, the time taken is arbitrary. The distance from the dock depends on the time taken, and since there is no specific time provided, we cannot determine the exact distance from the dock when the boy reaches the bow.