Internal energy of a decaying atom (simple energy problem)

[mnphys]

This is my first time here. Sorry if the formatting is a bit off. This is one of a few questions I am having difficulty with. I am not asking for any answers; and I have definitely done the work - multiple times - but have not been able to find the correct answer.

Q) A uranium-238 atom can break up into a thorium-234 atom and a particle called an alpha particle, α -4. The numbers indicate the inertias of the atoms and the alpha particle in atomic mass units (1 amu = 1.66 × 10⁻²⁷ kg ). When an uranium atom initially at rest breaks up, the thorium atom is observed to recoil with an x component of velocity of -2.9 × 10⁵ m/s.

- How much of the uranium atom's internal energy is released in the breakup? (Express your answer to three significant digits and include the appropriate units.)

Relevant Equations: k = 1/2mv²

- I found the velocity of the object to be -2.9 x 10⁵ m/s (given)
- I found the mass of the thorium atom to be 234 * 1.66 x 10^-27 kg = 388.44 x 10^-27 kg
- I found v² to be (-2.9 x 10⁵ m/s)² = 8.41 x 10¹⁰ m²/s²
- Multiply: 1/2 * 388.44 x 10^-27 kg * 8.41 x 10¹⁰ m²/s² = 1633.3902 x 10^-17 joules
- Reduce this to three significant digits: 163 x 10^-16 joules

Where did I go wrong? The program says that answer is not correct, but I went over it several times and can't find my mistake. Any help is appreciated. Please do not give me the answer. I doubt anyone here would, but that would violate my conscience (among other things!)

 

[SteamKing]

This is my first time here. Sorry if the formatting is a bit off. This is one of a few questions I am having difficulty with. I am not asking for any answers; and I have definitely done the work - multiple times - but have not been able to find the correct answer.

Q) A uranium-238 atom can break up into a thorium-234 atom and a particle called an alpha particle, α -4. The numbers indicate the inertias of the atoms and the alpha particle in atomic mass units (1 amu = 1.66 × 10⁻²⁷ kg ). When an uranium atom initially at rest breaks up, the thorium atom is observed to recoil with an x component of velocity of -2.9 × 10⁵ m/s.

- How much of the uranium atom's internal energy is released in the breakup? (Express your answer to three significant digits and include the appropriate units.)

Relevant Equations: k = 1/2mv²

- I found the velocity of the object to be -2.9 x 10⁵ m/s (given)
- I found the mass of the thorium atom to be 234 * 1.66 x 10^-27 kg = 388.44 x 10^-27 kg
- I found v² to be (-2.9 x 10⁵ m/s)² = 8.41 x 10¹⁰ m²/s²
- Multiply: 1/2 * 388.44 x 10^-27 kg * 8.41 x 10¹⁰ m²/s² = 1633.3902 x 10^-17 joules
- Reduce this to three significant digits: 163 x 10^-16 joules

Where did I go wrong? The program says that answer is not correct, but I went over it several times and can't find my mistake. Any help is appreciated. Please do not give me the answer. I doubt anyone here would, but that would violate my conscience (among other things!)

First of all, you should use the HW template when posting HW problems at PF. That's one of the rules.

Second, v² has units of (m/s)², not joules.

Third, although you are given the recoil velocity of the Th-234 nucleus, what happens to the alpha particle? Does it just sit there, after the U-238 nucleus has emitted it?

 

[mnphys]

Sorry, I thought I was using the template? I posted the question first, then the equations, then the attempt. If I didn't follow the rules I apologize.

I know nothing about the alpha particle's speed; all the information I was given was posted there. I thought it odd that they didn't mention anything but wasn't sure what to do about it.

 

[SteamKing]

Sorry, I thought I was using the template? I posted the question first, then the equations, then the attempt. If I didn't follow the rules I apologize.

When you hit the Create Thread button in the HW forums, the HW template with items 1., 2., and 3. appears in the editing box. Don't delete this template. Fill it out with your HW question, any relevant formulas or equations, and your attempts at solutions.

I know nothing about the alpha particle's speed; all the information I was given was posted there. I thought it odd that they didn't mention anything but wasn't sure what to do about it.

You're not given the speed of the alpha particle, but you are told that the U-238 nucleus was at rest before the alpha was emitted, and you are given the speed at which the Th-234 nucleus recoils. Using the Conservation of Momentum, you should be able to figure out the speed of the alpha.

 

[mnphys]

Thank you! I will be sure to follow that template if I have to ask any more questions. And your help was critical in solving this problem; given the mass of both particles and the velocity of one particle it's a simple matter of finding the velocity of the second particle and then calculating the kinetic energy of both particles, followed by adding them together.

So in assuming that the second particle was at rest, I made a critical error that violated the conservation of momentum. Lesson learned!

 

[SteamKing]

Glad everything worked out.