A child in a boat throws a 6.00 kg package out horizontally with a speed of 12.0 m/s, Fig. 7-28. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 25.0 kg and that of the boat is 50.0 kg.
Magnitude ___m/sDirection
45
To find the velocity of the boat immediately after the child throws the package, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before the package is thrown is equal to the total momentum after the package is thrown.
The momentum of an object is calculated as the product of its mass and velocity:
Momentum = mass * velocity
Before the package is thrown, both the child and the boat are at rest. Therefore, their initial momentum is zero:
Total initial momentum = momentum of child + momentum of boat = 0 + 0 = 0
After the package is thrown, the momentum of the child and the boat is in opposite directions. Let's assume that the velocity of the boat after the package is thrown is v (magnitude) and in the opposite direction to the thrown package's velocity.
The magnitude of the momentum of the child is given by:
Momentum of child = mass of child * velocity of child
= 25.0 kg * 12.0 m/s
= 300 kg⋅m/s
The magnitude of the momentum of the package is given by:
Momentum of package = mass of package * velocity of package
= 6.00 kg * 12.0 m/s
= 72.0 kg⋅m/s
According to the conservation of linear momentum, the total momentum after the package is thrown is equal to the sum of the momenta of the child, the boat, and the package:
Total final momentum = momentum of child + momentum of boat + momentum of package
Since the momenta of the child and the package are in opposite directions, their magnitudes should be subtracted:
Total final momentum = 300 kg⋅m/s - 72.0 kg⋅m/s
= 228 kg⋅m/s
Since the final momentum is equal to the momentum of the boat alone (as the child and the package are no longer on the boat), we can calculate the velocity of the boat using its mass:
Total final momentum = mass of boat * velocity of boat
Rearranging the equation, we can solve for the velocity of the boat:
Velocity of boat = Total final momentum / mass of boat
= 228 kg⋅m/s / 50.0 kg
= 4.56 m/s
Therefore, the magnitude of the velocity of the boat immediately after the child throws the package is 4.56 m/s. The direction of the velocity is opposite to the direction in which the package was thrown.