(a) Use the binomial expansion to show that Eqs. 9a and 10a...

(a) Use the binomial expansion to show that Eqs. 9a and 10a become essentially the same for small relative velocity between source and observer. (b) What percent error would result if Eq. 10 $\mathrm{a}$ were used instead of Eq. 9 a for a relative velocity of 18.0 $\mathrm{m} / \mathrm{s} ?$ $$f^{\prime}=\frac{f}{\left(1-\frac{v_{\text { source }}}{v_{\text { sond }}}\right)}$$ $$\left[ \begin{array}{l}{\text { source moving toward }} \\ {\text { stationary observer }}\end{array}\right]$$ $$f^{\prime}=\left(1+\frac{v_{\mathrm{obs}}}{v_{\mathrm{snd}}}\right) f$$ $$\left[ \begin{array}{l}{\text { observer moving toward }} \\ {\text { stationary source }}\end{array}\right]$$

(a) Use the binomial expansion to show that Eqs. 9a and
10a become essentially the same for small relative velocity between source and observer. (b) What percent error would result if Eq. 10 $\mathrm{a}$ were used instead of Eq. 9 a for a relative velocity of 18.0 $\mathrm{m} / \mathrm{s} ?$
$$f^{\prime}=\frac{f}{\left(1-\frac{v_{\text { source }}}{v_{\text { sond }}}\right)}$$ $$\left[ \begin{array}{l}{\text { source moving toward }} \\ {\text { stationary observer }}\end{array}\right]$$
$$f^{\prime}=\left(1+\frac{v_{\mathrm{obs}}}{v_{\mathrm{snd}}}\right) f$$
$$\left[ \begin{array}{l}{\text { observer moving toward }} \\ {\text { stationary source }}\end{array}\right]$$

Key Concepts

Binomial Expansion

The binomial expansion is a mathematical method used to express functions of the form (1 + x)^n as a series, especially useful when x is small. In physics, this approximation allows complex expressions to be simplified by retaining only the first few terms, thus facilitating the comparison of different formulations of an equation in the limit of small parameters.

Small Relative Velocity Approximation

The small relative velocity approximation assumes that the speed of the moving object is much less than the speed of the wave (such as sound). This assumption justifies the truncation of higher-order terms in the binomial expansion, making it possible to show that different Doppler effect formulas reduce to essentially the same expression under low-speed conditions.

Doppler Effect

The Doppler Effect is the phenomenon in which the frequency of a wave, such as sound or light, changes due to the motion of the source relative to an observer. This concept is central to understanding how the observed frequency varies when either the source or the observer is moving, which is the foundation behind the equations being compared in the problem.

Percentage Error Analysis

Percentage error analysis is a method used to quantify the deviation between an approximate value and the true value by expressing the difference as a percentage. In the context of the Doppler effect equations, it allows one to evaluate the accuracy of using one approximate formula over another when the relative velocity is finite, thereby assessing the suitability of the approximation for practical applications.

Transcript

00:02 Set a use the binomial expansion to show that equation 9a and 10a become essentially the same for small relative velocities between source and observer and b what percent error would result if 10 a were used instead of 9 a for a relative velocity of 18 meters per second? okay so we'll write the the relative velocity v -s -s of s which is the source here velocity this is going to be used in part b and is equal to 18 .0 meters per second.

00:38 Okay.

00:40 And then indirectly, we're also using the speed of sound, which we're going to call here c.

00:45 So the speed of sound is in this question we're going to refer to us as c is 343.

00:53 Let's write that down where we can see it, 343 meters per second.

01:03 Okay.

01:05 So for part a, we are asked to assume that v sub s is much less than c okay f prime of s the frequency of the observer of the source moving is equal to the frequency times one over one minus the velocity of the source divided by c v sub s divided by c the speed of sound which is equal to f times the inverse of one minus v .s over c.

01:46 So we're just arranging the equation.

01:49 Okay.

01:51 Well, when v .s is much less than c, this is approximately equal to f times 1 plus v.

02:01 Sub s over c, which is, that expression is the same as f prime of the observer moving.

02:13 So we just showed that they're equivalent for small values of v .s relative to c.

02:17 That was what we were asked to do for part a, so proven...

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