Modern Physics 157

Chapter 6: Modern Physics 157

The general decay curve for a radioactive sample relating the number of nuclei present at a given time to the original number of nuclei is - λ exponential. The expression is t NN =

o e , where N is the original number of nuclei, N o is the number of nuclei at time t, and e is the base of the nat- ural logarithm.

Nuclear reactions

Nuclear fission occurs when a heavy nucleus splits into two nearly equal

92 size nuclei. The reaction for uranium 235 is 1

0 n + 92 U " 56 Ba + 36 Kr 3 + 0 n . The total rest mass of the products is less than the original rest mass of the original uranium by 220 MeV. This is an enormous amount of energy compared to energy releases in chemical processes and when considering that a relatively modest piece of uranium has so very many nuclei. Nuclear fusion occurs when light nuclei are combined to form a heavier nucleus. The sun is powered by nuclear fusion.

The binding energy is related to stability. When the mass energy of the parent nucleus is greater than the total mass energy of the decay products, spontaneous decay will take place. If the decay products have a greater total mass energy than the parent nucleus, additional energy is necessary for the reaction to occur. Energy is released when light nuclei combine (fusion) and when heavy nuclei split (fission).

Chapter Checkout

Q&A

1. The mother ship is traveling through space at a constant velocity of 0.9c, relative to an observer on a nearby moon. A shuttle is launched from the mother ship at 0.5c relative to the mother ship. What speed does the shuttle have with respect to the observer on the moon?

2. A muon travels through the upper atmosphere with a velocity of 0.85c relative to the ground. The lifetime of a muon at rest (or, in its own reference frame) is 1 µs. The muon is at the top a mountain which is 10,000 m tall, measured in its rest frame. (a) From the muon’s perspective, how high is the mountain? How long would it take the muon to travel the length of the mountain? (b) For an observer on the mountain, what is the lifetime of the muon? How far does the observer see the muon travel in that time? (c) How would you reconcile the experiences of the muon and the observer? Does

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3. A spaceship captain orders his crew to accelerate their ship to the speed of light. (a) How much energy is required to increase the speed

of the ship from 0.1c to 0.2c, if the ship has a rest mass of 10 5 kg? (b) How much energy is required to accelerate between 0.9c and 0.95c?

(c) From 0.98 c to 0.99c? What would you tell the captain?

4. (a) Calculate your wavelength and momentum, if you have a mass of 80 kg and are riding a skateboard at 1 m/s. (b) Calculate the momentum of a photon, assuming it has a wavelength of 650 nm.

5. (a) Calculate the value of the Bohr radius. (b) Consider an ionized atom of helium, which would have one electron orbiting a nucleus with two protons (and two neutrons). What is the radius of the n = 1 state in this case? You will need to repeat Bohr’s calculation of the cen- tripetal force due to Coulomb’s law.

Answers: 1. 0.965 c 2a. 5268 m; 20 µs b. 1.9 µs; 484.5 m c. The muon

20 21 doesn’t reach the ground 3a. 1.4 × 10 22 J b. 8.1 × 10 J c. 1.9 × 10 J

4a. λ = 8.25 × 10 –27 m; p = 80 kg m/s b. 10 kg m/s 5a. 0.53 A

b. 0.265 A