A child in a boat throws a 5.40 kg package out horizontally with a speed of 10.0 m/s, Fig. 7-31. 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 40.0 kg.

To find the velocity of the boat immediately after the child throws the package, we can use the principle of conservation of momentum. According to this principle, the total momentum of a system before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.

The momentum of an object is calculated by multiplying its mass by its velocity: momentum = mass x velocity.

Let's denote the final velocity of the boat as v. Initially, both the child and the boat are at rest, so their total momentum is zero. After the child throws the package, the momentum of the system should also be zero.

The momentum of the child before throwing the package is given by: momentum_child = mass_child x velocity_child.

The momentum of the package is: momentum_package = mass_package x velocity_package.

The momentum of the boat after the package is thrown is: momentum_boat = mass_boat x velocity_boat.

Using the principle of conservation of momentum, we can write the equation:

momentum_child + momentum_package + momentum_boat = 0

(mass_child x 0) + (mass_package x velocity_package) + (mass_boat x v) = 0

Since the child initially at rest, the first term is zero.

mass_package x velocity_package + mass_boat x v = 0

Substituting the known values:

25.0 kg (mass_child) x 10.0 m/s (velocity_package) + 40.0 kg (mass_boat) x v = 0.

Now, we can rearrange the equation to solve for v:

25.0 kg x 10.0 m/s + 40.0 kg x v = 0

250 kg*m/s + 40.0 kg x v = 0

40.0 kg x v = -250 kg*m/s

Dividing both sides by 40.0 kg:

v = -250 kg*m/s / 40.0 kg = -6.25 m/s

Therefore, the velocity of the boat immediately after the child throws the package is -6.25 m/s, with the negative sign indicating that the boat moves in the opposite direction of the thrown package.