Question

A wave has a wavelength of 4.9 m and a velocity of 9.8 m/s. The medium through which this wave is traveling is then heated so that the velocity is doubled.

If the frequency remains constant, what is the wavelength of the heated wave?

2.5 m
9.8 m
14.7 m
19.6 m

Answer 1

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**.