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9. (11 points) The theory of relativity predicts that when an object moves at speeds close to the
speed of light, the object appears heavier. The apparent, or relativistic, mass m, of the ob ject
when it is moving at speed vis given by the formula
m=m01−v2
c2−1/2
where cis the speed of light and m0is the mass of the object when it is at rest.
(a) (8 points) Write the first four nonzero terms of the Taylor series for min terms of v. (Hint: You
may want to use the binomial series.)
Since mis given by
m=m01−v2
c2−1/2
,
and 0 ≤v < c, we may use the binomial series
(1 + x)p= 1 + px +p(p−1)
2! x2+p(p−1)(p−2)
3! x3+· · · ,
with x=−v2/c2and p=−1/2, to obtain:
m=m01 + 1
2c2v2+3
8c4v4+5
16c6v6+· · · .
(b) (3 points) The series you derived in part (a) converges for vin the interval [0, c). Interpret the
practical significance of this interval of convergence in the context of this problem (that is, as far
as the relativistic mass of an object is concerned.)
When an object’s speed (v) is less than the speed of light (c) the relativistic mass (m) of the
object is finite. In this case, the Taylor series for the relativistic mass (derived above) converges
to the actual relativistic mass.
University of Michigan Department of Mathematics Fall, 2005 Math 116 Exam 2 Problem 9 (relativity) Solution
