To solve the problem, we will use the concept of the Doppler effect, which describes how the frequency of a wave changes for an observer moving relative to the source of the wave. Here, we have two tuning forks with the same natural frequency, one moving towards the observer and the other moving away from the observer, and we need to find the speed of the tuning forks based on the beat frequency heard by the observer.
### Step-by-Step Solution:
1. **Identify the Given Values:**
- Natural frequency of each tuning fork, \( f = 340 \, \text{Hz} \)
- Beat frequency, \( f_b = 3 \, \text{Hz} \)
- Speed of sound in air, \( v = 340 \, \text{m/s} \)
2. **Understand the Doppler Effect:**
- For the tuning fork moving towards the observer, the observed frequency \( f_1 \) is given by:
\[
f_1 = f \frac{v}{v - v_s}
\]
- For the tuning fork moving away from the observer, the observed frequency \( f_2 \) is given by:
\[
f_2 = f \frac{v}{v + v_s}
\]
3. **Set Up the Beat Frequency Equation:**
- The beat frequency is the absolute difference between the two observed frequencies:
\[
f_b = |f_1 - f_2|
\]
- Substituting the expressions for \( f_1 \) and \( f_2 \):
\[
3 = \left| f \frac{v}{v - v_s} - f \frac{v}{v + v_s} \right|
\]
4. **Factor Out the Common Frequency:**
- Factor out \( f \) from the equation:
\[
3 = f \left( \frac{v}{v - v_s} - \frac{v}{v + v_s} \right)
\]
- This simplifies to:
\[
3 = 340 \left( \frac{v}{v - v_s} - \frac{v}{v + v_s} \right)
\]
5. **Combine the Fractions:**
- Combine the fractions on the right-hand side:
\[
\frac{v(v + v_s) - v(v - v_s)}{(v - v_s)(v + v_s)} = \frac{2v_s v}{(v - v_s)(v + v_s)}
\]
- Thus, the equation becomes:
\[
3 = 340 \cdot \frac{2v_s v}{(v - v_s)(v + v_s)}
\]
6. **Substituting Known Values:**
- Substitute \( v = 340 \, \text{m/s} \):
\[
3 = 340 \cdot \frac{2v_s \cdot 340}{(340 - v_s)(340 + v_s)}
\]
7. **Cross-Multiply and Simplify:**
- Cross-multiply to eliminate the fraction:
\[
3(v - v_s)(v + v_s) = 680 v_s v
\]
- Substitute \( v = 340 \):
\[
3(340^2 - v_s^2) = 680 \cdot 340 \cdot v_s
\]
8. **Solve for \( v_s \):**
- Rearranging gives:
\[
1020 v_s^2 + 680 \cdot 340 v_s - 3 \cdot 340^2 = 0
\]
- This is a quadratic equation in \( v_s \). Using the quadratic formula:
\[
v_s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- Here, \( a = 1020 \), \( b = 680 \cdot 340 \), and \( c = -3 \cdot 340^2 \).
9. **Calculate the Values:**
- After performing the calculations, we find:
\[
v_s = 1.5 \, \text{m/s}
\]
### Final Answer:
The speed of the tuning forks is \( v_s = 1.5 \, \text{m/s} \).
