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An electron is accelerated from rest through a potential difference that has a magnitude of 2.90 × 107 V. The mass of the electron is 9.11 × 10-31 kg, and the negative charge of the electron has a magnitude of 1.60 × 10-19 C. (a) What is the relativistic kinetic energy (in joules) of the electron? (b) What is the speed of the electron? Express your answer as a multiple of c, the speed of light in a vacuum.
to find the speed,
so take relavitic KE, divide by restmass, then divide by speed of light.
Now ignore the -1 in the gamma, look at it later. If v is approaching c, it can be ignored.
square both sides (ignoring the -1)
solve for v in terms of c. If v/c is within .5 or greater, use that v, it is pretty accurate. If v/c is greater than .5, you have to consider the -1 in gamma. You can do that by expanding the gamma with the binomial expansion, and there are programs to do that, and I am told calculators to do it. I would not relish doing more than a few terms by hand.