A sled of mass 9.0 kg is sliding across a frozen lake with a speed of 5.0 m/s. A package of mass 4.0 kg is dropped onto it from above. Assuming there is no friction between the sled and the lake, what is the new speed of the sled?

By the law of conservation of momentum,

m1v1 = m2v2
9 * 5 = (9+4) v2
v2 = 45/13

To determine the new speed of the sled after the package is dropped onto it, we can use the principle of conservation of momentum.

The total momentum before the package is dropped is equal to the total momentum after the package is dropped. Momentum is calculated as the product of mass and velocity.

Before the package is dropped, the sled has a mass of 9.0 kg and a speed of 5.0 m/s. Therefore, the momentum of the sled before the package is dropped is given by:

Momentum of sled before = mass of sled × velocity of sled
= 9.0 kg × 5.0 m/s
= 45 kg·m/s

The package has a mass of 4.0 kg and it is initially at rest, so its momentum before is zero.

Therefore, the total momentum before the package is dropped is equal to the momentum of the sled before:

Total momentum before = Momentum of sled before + Momentum of package before
= 45 kg·m/s + 0 kg·m/s
= 45 kg·m/s

After the package is dropped, the two masses (sled and package) move together with a common final velocity, which we need to find.

Let's assume the final velocity of the combined sled and package is v.

The momentum of the sled after the package is dropped is given by:

Momentum of sled after = mass of sled × velocity of sled after
= 9.0 kg × v
= 9v kg·m/s

The momentum of the package after it is dropped is given by:

Momentum of package after = mass of package × velocity of package after
= 4.0 kg × v
= 4v kg·m/s

The total momentum after the package is dropped is equal to the sum of the momenta of the sled and the package:

Total momentum after = Momentum of sled after + Momentum of package after
= 9v kg·m/s + 4v kg·m/s
= 13v kg·m/s

According to the principle of conservation of momentum, the total momentum before the package is dropped is equal to the total momentum after the package is dropped:

Total momentum before = Total momentum after
45 kg·m/s = 13v kg·m/s

Solving for v:

v = 45 kg·m/s / 13 kg·m/s

v ≈ 3.46 m/s

Therefore, the new speed of the sled after the package is dropped is approximately 3.46 m/s.