To solve for the new wavelength of the wave after the velocity is doubled, we can use the wave equation:
\[ v = f \lambda \]
where:
- \( v \) is the wave velocity,
- \( f \) is the frequency, and
- \( \lambda \) is the wavelength.
Initially, we have:
- Wavelength (\( \lambda \)) = 4.9 m,
- Velocity (\( v \)) = 9.8 m/s.
First, we can find the frequency:
\[ f = \frac{v}{\lambda} = \frac{9.8 , \text{m/s}}{4.9 , \text{m}} = 2 , \text{Hz}. \]
Now, when the velocity is doubled, the new velocity (\( v' \)) is:
\[ v' = 2 \times 9.8 , \text{m/s} = 19.6 , \text{m/s}. \]
Since the frequency remains constant (\( f = 2 , \text{Hz} \)), we can solve for the new wavelength (\( \lambda' \)) using the wave equation again:
\[ v' = f \lambda'. \]
Substituting the new values:
\[ 19.6 , \text{m/s} = 2 , \text{Hz} \times \lambda'. \]
To find \( \lambda' \):
\[ \lambda' = \frac{19.6 , \text{m/s}}{2 , \text{Hz}} = 9.8 , \text{m}. \]
Thus, the new wavelength of the heated wave is \( 9.8 , \text{m} \).
Final answer: 9.8 m.