Special theory of relativity: velocity of airplane

[salmayoussef]

Homework Statement

If a clock in an airplane is found to slow down by 5 parts in 10¹³, (i.e. Δt/Δt_o = 1 + 5x10^-13), at what speed is the airplane travelling? (Hint: You may need to use the binomial expansion for γ.)

To be honest, I'm really confused about what this question's telling me. What's in the brackets?

Homework Equations

γ = 1/√(1-v²/c²)
That's the equation I used.

The Attempt at a Solution

When I rearranged the equation to find the velocity, I got v = √((1-(1/γ)²)(c²))

I subbed in 1 + 1.5x10^-13 for γ but I keep getting 0 as my velocity and the answer's supposed to be 300 m/s. Am I even doing this right? Thanks in advance! I appreciate any help you can give me!

 

[dauto]

1st why did you sub 1 + 1.5x10^-13 for γ instead of 1 + 5x10^-13, as described in the problem.

2nd yes your method is correct but either you're misusing the calculator or the calculator can't handle the calculation. Yes, calculators don't always provide correct answers. That's why the problem said "You may need to use the binomial expansion for γ" by what they mean a Taylor expansion.

 

[salmayoussef]

why did you sub 1 + 1.5x10^-13 for γ

Oops! Sorry! I meant to write 5x10^-13. I'm afraid I don't really know what a Taylor expansion is... How does this work?

 

[dauto]

Taylor's expansion is very important and useful but it takes more than a few lines to explain it. You should definitely learn it, but for right now I would try using a better calculator. Use the calculator that comes with Windows. It's very good.

 

[vela]

The binomial expansion says that
$$(1+x)^m = 1 + \frac{m}{1!} x + \frac{m(m-1)}{2!}x^2 + \frac{m(m-1)(m-2)}{3!}x^3 + \cdots.$$ In particular, for $m=-1/2$, you get
$$(1+x)^{-1/2} \cong 1 - \frac 12 x.$$ Therefore, you have
$$\gamma = \left[ 1 + \left(-\frac{v^2}{c^2}\right)\right]^{-1/2} \cong 1 + \frac 12 \frac{v^2}{c^2}.$$