Question

An ultrasound signal of frequency 50 KHz is sent vertically down into a medium. The signal gets reflected from a depth of 25 mm and returns to source 0.00005 seconds after it is emitted. The wavelength of the ultrasound signal in that medium is ............... cm. 
 

Hint:

The wavelength of a wave is given by \( \lambda = \frac{v}{f} \), where \( v \) is the speed and \( f \) is the frequency.

Answer 1

Correct Answer: 1.99 - 2.01

Step 1: Understanding the formula for wave speed. 
The wave speed \( v \) of the ultrasound in the medium can be found using the formula: \[ v = \frac{\text{distance}}{\text{time}} \] The signal travels to the depth and back, so the total distance is \( 2 \times 25 \, \text{mm} = 50 \, \text{mm} = 0.05 \, \text{m} \).

Step 2: Calculating the wave speed. 
The time taken for the wave to travel this distance is \( 0.00005 \) seconds. Therefore, the wave speed is: \[ v = \frac{0.05 \, \text{m}}{0.00005 \, \text{s}} = 1000 \, \text{m/s} \]

Step 3: Finding the wavelength. 
The wavelength \( \lambda \) is related to the wave speed \( v \) and frequency \( f \) by the formula: \[ v = f \lambda \] Substitute the known values: \[ 1000 \, \text{m/s} = 50,000 \, \text{Hz} \times \lambda \] Solving for \( \lambda \): \[ \lambda = \frac{1000}{50,000} = 0.02 \, \text{m} = 2 \, \text{cm} \]

Step 4: Conclusion. 
Thus, the wavelength of the ultrasound signal in that medium is 2 cm.