Physics - Momentum - Please please help!!!!!!!!!!!!!!!!!!!?

Question

1. A child throws a 5.40 kg package horizontally from a boat with a speed of 10 m/s. Calculate the resulting velocity of the boat, assuming it was initially at rest. The mass of the child is 25 kg and the boat is 55 kg.

2. An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. what will be the speed of this recoiling nucleus if the speed of the alpha particle is 3.8 x 10^5 m/s? Assume the recoiling nucleus has mass 57 times greater than that of the alpha particle.

The answers from the back of the book are 1) -0.675 m/s
and 2) -6.7 x 10^3 m/s.

I don't understand how to get these answers so please show your work and explain. Thanks!

Answer 1

The same principle is at work in both these problems: momentum is conserved (unchanged). They both start with M = 0, so after the action takes place the momentum must still be 0......

1) The momentum equation is Mp + Mb = 0 → 5.4*10 + (25+55)*Vb = 0. when you solve this for Vb, you get Vb = -.675 m/s

2) Here we aren't told the masses, but we are told their ratio, which is really all we need: Mn*Vn = - Ma*Va → Vn = -(Ma/Mn)*Va = -(1/57)*3.8E5 = -6.67*10³ m/s

Hope this helps; if not, email me

Answer 2

We will use conservation of momentum. In the first problem, the initial momentum is zero. That is because nothing is moving to begin with.
Now, the final momentum is also zero, and it is the sum of the momentum of the package and the "boat plus child".
So, the momentum of the flying package is 5.4*10 = 54 kg m/s
The momentum of the child+boat must then be -54 kg m/s
Momentum is the mass times the speed. We know the mass (55+25), so we solve for speed S:
(55+25)*S = -54
or S = -54/(55+25) which sounds consistent with -.675

To solve the second problem, you do essentially the same thing, but you will need to make up a "mass" for the alpha particle and the nucleus. Make up 1 and 57 for the masses, and you'll see that the units don't matter anyway.