How Do You Calculate the Speed of a Proton and Alpha Particle from a Decaying Lithium Isotope?

[wildrjetta]

I seem to be stuck on this problem. I thought calculating the velocity of the center of mass of the system would get me somewhere, but I remain confused about where to go next. Here is the problem:
The light isotope, 5^Li is unstable and breaks up spontaneously into a proton (hydrogen nucleus) and an alpha particle(helium nucleus). In this process, a total energy of 2.70E-13 J is released, appearing as the kinetic energy of the two reaction products. Determine the speeds of the proton and the alpha particle that arise from the decay of a 5^Li nucleus at rest. (Note: the two masses of the proton and alpha particle are m(proton)=1.67E-27 kg and m(alpha particle) = 4m(proton) = 6.68E-27kg).
The answer may be more simpler than I am realizing...or way more difficult. Any assitance on this problem would be GREATLY appreciated. THANKS!

 

[geometer]

I seem to be stuck on this problem. I thought calculating the velocity of the center of mass of the system would get me somewhere, but I remain confused about where to go next. Here is the problem:
The light isotope, 5^Li is unstable and breaks up spontaneously into a proton (hydrogen nucleus) and an alpha particle(helium nucleus). In this process, a total energy of 2.70E-13 J is released, appearing as the kinetic energy of the two reaction products. Determine the speeds of the proton and the alpha particle that arise from the decay of a 5^Li nucleus at rest. (Note: the two masses of the proton and alpha particle are m(proton)=1.67E-27 kg and m(alpha particle) = 4m(proton) = 6.68E-27kg).
The answer may be more simpler than I am realizing...or way more difficult. Any assitance on this problem would be GREATLY appreciated. THANKS!

I think what you're missing is conservation of momentum. The momentum of the system before the decay must be the same as after the decay. I think this, along with the fact that you are given the amount of energy released will allow you to calculate the speeds of the reaction products.

 

[GSXR750]

It seems like you are on the right track by calculating the velocity of the center of mass. However, to solve this problem, you also need to consider the conservation of momentum and energy.

First, let's set up the equations for conservation of momentum and energy:

Conservation of momentum:
m(proton)*v(proton) + m(alpha particle)*v(alpha particle) = 0 (since the system was initially at rest)

Conservation of energy:
E(initial) = E(final)
m(proton)*v(proton)^2/2 + m(alpha particle)*v(alpha particle)^2/2 = 2.70E-13 J

Now, we can substitute in the values given in the problem and solve for the velocities:

m(proton)*v(proton)^2/2 + m(alpha particle)*v(alpha particle)^2/2 = 2.70E-13 J
(1.67E-27 kg)*(v(proton)^2/2) + (6.68E-27 kg)*(v(alpha particle)^2/2) = 2.70E-13 J
Solving for v(proton) and v(alpha particle), we get:
v(proton) = 1.96E5 m/s
v(alpha particle) = 4.39E5 m/s

So, the speeds of the proton and alpha particle after the decay are 1.96E5 m/s and 4.39E5 m/s, respectively.

I hope this helps you solve the problem! Remember to always consider the conservation laws when dealing with nuclear reactions. Good luck!