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Energy from Matter Help (page 3)

By — McGraw-Hill Professional
Updated on Sep 5, 2011

Problem 2

We know that 1 W = 1 J/s. How long would the energy produced in the preceding matter-antimatter reaction, if it could be controlled and harnessed, illuminate a 100-W light bulb?

Solution 2

Divide the amount of energy in joules by the wattage of the bulb in joules per second. We know this will work because, in terms of units,

J/W = J/(J/s) = J · (s/J) = s

The joules cancel out. Note also that the small dot (·) is used to represent multiplication when dealing with units, as opposed to the slanted cross (×) that is customarily used with numerals. Getting down to the actual numbers, let P be the power consumed by the bulb (100 W), let t s be the number of seconds the 100-W light bulb will burn, and let E be the total energy produced by the matter-antimatter reaction, 1.80 × 10 17 J. Thus

t s = E/P

= 1.80 × 10 17 /100

= 1.80 × 10 17 /10 2

= 1.80 × 10 15 s

This is a long time, but how long in terms of, say, years? There are 60.0 seconds in a minute, 60.0 minutes in an hour, 24.0 hours in a day, and, on average, 365.25 days in a year. This makes 31,557,600, or 3.15576 × 10 7 , seconds in a year. Let t yr be the time in years that the light bulb burns. Then

t yr = t s /(3.15576 × 10 7 )

= (1.80 × 10 15 )/(3.15576 × 10 7 )

= 0.570 × 10 8

= 5.70 × 10 7 yr

This is 57.0 million years, rounded to three significant figures (the nearest 100,000 years), which is all the accuracy to which we are entitled based on the input data.

Problem 3

Suppose that the amount of matter in the preceding two problems is doubled to 2.00 kg but the amount of antimatter remains 1.00 kg. How much energy will be liberated? Will there be any matter or antimatter left over?

Solution 3

The amount of liberated energy will be the same as in the examples shown by the preceding two problems: 1.80 × 10 17 J. There will be 1.00 kg of matter left over (the difference between the masses). However, assuming that the encounter produces an explosion, the matter won’t remain in the form of a brick. It will be scattered throughout millions of cubic kilometers of space.

Practice problems of these concepts can be found at: Particles of Matter Practice Test

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