Approximating an expression with the binomial expansion

[b2386]

$$f_r=(\frac{1+v}{1-v})f_i$$

For an automobile moving at speed v that is a small fraction of the speed of light, assume that the fractional change in frequency of reflected radar is small. Under this assumption, use the first two terms of the bionomial expansion

$$(1-x)^n\approx{1-nz \mbox{for} |z| \ll{1}$$

to show that the fractional change of frequency is given by the approximate expression

$$\frac{\Delta{f}}{f}\approx{2v}$$



So far, I have

\frac{\Delta{f}}{f}=\frac{\frac{1+v}{1-v}\Delta{f_i}}{\frac{1+v}{1-v}f_i

Now, does the binomial expansion allow this?: $$\frac{1+v}{1-v}\approx{(1+v)(1+v)}$$

Is this where this problem wants me to go?

 

[b2386]

Can someone also please show me the errors in my two non-latex equations? They don't seem to want to play nicely.

 

[kyle8921]

Yes, you are on the right track. The binomial expansion can be used to approximate expressions when the value inside the parentheses is small. In this case, we can use it to approximate the expression $\frac{1+v}{1-v}$ as $(1+v)(1+v)$ since the value of $v$ is small compared to 1. This allows us to rewrite the original expression as:

\frac{\Delta{f}}{f}=\frac{(1+v)(1+v)\Delta{f_i}}{(1+v)(1+v)f_i}

Now, we can use the binomial expansion to approximate $(1+v)^2\approx 1+2v$. This gives us:

\frac{\Delta{f}}{f}\approx\frac{(1+2v)\Delta{f_i}}{(1+2v)f_i}

Simplifying further, we get:

\frac{\Delta{f}}{f}\approx\frac{\Delta{f_i}}{f_i}+\frac{2v\Delta{f_i}}{(1+2v)f_i}

Since we are assuming that the fractional change in frequency is small, we can neglect the first term and only consider the second term. This gives us the approximate expression:

\frac{\Delta{f}}{f}\approx\frac{2v\Delta{f_i}}{(1+2v)f_i}

Finally, we can substitute in the original expression for $\Delta{f_i}$, which is the change in frequency of the incident radar signal. This gives us the final approximate expression:

\frac{\Delta{f}}{f}\approx\frac{2v}{1-v}\cdot\frac{f_i}{f_i}

Simplifying further, we get:

\frac{\Delta{f}}{f}\approx\frac{2v}{1-v}

This shows that the fractional change in frequency is approximately equal to $2v$, which is the speed of the object relative to the speed of light. This approximation can be used when the speed of the object is a small fraction of the speed of light.