a stationary speed gun emits a microwave beam at 2.41×10^10 hz. it reflects off a pitched baseball and…

Explanation:

Step1: Recall the Doppler - effect formula for a moving object reflecting a wave

The Doppler - shift formula for a wave emitted by a stationary source and reflected by a moving object (in this case, the baseball) is $\Delta f=\frac{2v}{c}f_0$, where $\Delta f$ is the change in frequency, $v$ is the speed of the moving object, $c = 3\times10^{8}\ m/s$ is the speed of light (speed of the microwave), and $f_0$ is the original frequency of the emitted wave.

Step2: Rearrange the formula to solve for $v$

Starting from $\Delta f=\frac{2v}{c}f_0$, we can solve for $v$. First, multiply both sides of the equation by $c$: $c\Delta f = 2vf_0$. Then, divide both sides by $2f_0$ to get $v=\frac{c\Delta f}{2f_0}$.

Step3: Substitute the given values

We are given that $f_0 = 2.41\times 10^{10}\ Hz$, $\Delta f=3190\ Hz$, and $c = 3\times 10^{8}\ m/s$. Substituting these values into the formula $v=\frac{c\Delta f}{2f_0}$, we have $v=\frac{3\times 10^{8}\times3190}{2\times2.41\times 10^{10}}$. First, calculate the numerator: $3\times 10^{8}\times3190 = 3\times3190\times10^{8}=9570\times 10^{8}=9.57\times 10^{11}$. Then, calculate the denominator: $2\times2.41\times 10^{10}=4.82\times 10^{10}$. Now, divide the numerator by the denominator: $v=\frac{9.57\times 10^{11}}{4.82\times 10^{10}}$. Using the rule of exponents $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we get $v=\frac{9.57}{4.82}\times10^{11 - 10}\approx19.9\ m/s$.

Answer:

$19.9\ m/s$