Given that ` y =A sin [( ( 2pi )/( lambda ) ( ct -x))] ` ,where y and x are measured in metres. Which of the following statement is true?

Options

A. The unit of `lambda` is same as that of x and A.
B. The unit of `lambda` is same as that of x but may not be same as that of `A`.
C. The unit of `c` is same as that of ` 2 pi // lambda`.
D. The unit of `( c t - x)` is same as that of ` 2 pi // lambda`.

Explanation

To solve the problem, we need to analyze the equation given:

\[ y = A \sin\left(\frac{2\pi}{\lambda}(ct - x)\right) \]

where \( y \) and \( x \) are measured in meters. We need to determine which statement regarding the units of \( \lambda \), \( A \), \( c \), \( t \), and \( x \) is true.

### Step-by-Step Solution:

1. **Identify the components of the equation**:
- The equation involves \( y \), \( A \), \( \lambda \), \( c \), \( t \), and \( x \).
- Here, \( y \) is the displacement, \( A \) is the amplitude, \( \lambda \) is the wavelength, \( c \) is the speed of the wave, \( t \) is time, and \( x \) is the position.

2. **Determine the dimensions of each component**:
- \( y \) and \( x \) are both measured in meters (m).
- The amplitude \( A \) is also a measure of displacement, so it has the same unit as \( y \), which is meters (m).
- The wavelength \( \lambda \) is also a measure of distance, hence its unit is meters (m).
- The speed \( c \) has units of meters per second (m/s).
- Time \( t \) is measured in seconds (s).

3. **Analyze the argument of the sine function**:
- The argument of the sine function must be dimensionless. Therefore, the term \(\frac{2\pi}{\lambda}(ct - x)\) must be dimensionless.
- This means that \( ct - x \) must have the same units as \( \lambda \) (which is meters).
- Since \( ct \) has units of \( c \times t = \text{(m/s)} \times \text{s} = \text{m} \), it has the same unit as \( x \) (which is meters).

4. **Equate the units**:
- Since \( ct \) and \( x \) are both in meters, it follows that \( ct - x \) is also in meters.
- Thus, for the expression \(\frac{2\pi}{\lambda}(ct - x)\) to be dimensionless, \( \lambda \) must also be in meters.

5. **Conclusion about the units**:
- From the analysis, we conclude that \( \lambda \), \( x \), and \( A \) all have the same unit (meters).
- However, \( c \) (speed) has units of m/s, which is different from the units of \( \lambda \), \( A \), and \( x \).

### Final Statement:
The correct statement is that the unit of \( \lambda \) is the same as that of \( x \) and \( A \), but it may not be the same as that of \( c \).