The forward and backward movement of an air molecule in still air during the transmission of a middle C note is given by $$x = A\sin (2272t)$$ where $t$ is the time in seconds and $A$ is the maximum displacement of the molecule. $A$ is related to the loudness of the sound.
Q: How many times does the molecule have a velocity of zero in 1 second?
So I have answered this question in two ways but they give me different answers and I would like to know why one works and the other one doesn't.
Method 1:
- Derive the function to get $x'=2272A\cos(2272t)$
- Calculated the frequency and got a frequency of approx 361
- since cosine has two zeros per period: $361\cdot 2 = 723$ times molecule has zero velocity in 1 second.
Method 2:
- Derived function as method 1.
- since cosine has zeros at every $n\pi/2$, equated $\cos(2272t)=n\pi/2$
- subbed $t=1$ and solved for $n$ and got $1446$ which is double method 1's answer.
The correct answer is method 1 but could anyone clarify why method 2 does not give me the same answer?
The function $\cos(x)$ has a zero at $x=\frac{\pi}{2}+n\pi$ for any integer $n$. There are two zeros per period $T=2\pi$. In saying that the cosine has a zeros at every $n\pi/2$, you doubled the number of such zeros.