Given that` y = A sin [((2 pi )/(lambda)(ct - x))]` where` y` and `x` are measured in metres ,Which of the following statements 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 not of ` A`
C. The unit of c is same as that of `(2 pi)/(lambda)`
D. The unit of `(a -x )` is same as that of `(2 pi)/(lambda)`

Explanation

To analyze the given equation \( y = A \sin\left(\frac{2\pi}{\lambda}(ct - x)\right) \) and determine which statement is true, we need to break down the components of the equation and their respective units.

### Step-by-Step Solution:

1. **Identify the Variables:**
- \( y \): Displacement (measured in meters)
- \( A \): Amplitude (also measured in meters)
- \( \lambda \): Wavelength (measured in meters)
- \( c \): Speed (measured in meters per second)
- \( t \): Time (measured in seconds)
- \( x \): Position (measured in meters)

2. **Analyze the Units:**
- Since \( y \) and \( A \) are both measured in meters, we can conclude that:
\[ [y] = [A] = L \quad (\text{where } L \text{ represents meters}) \]
- The unit of \( x \) is also meters:
\[ [x] = L \]
- The unit of \( \lambda \) (wavelength) is also meters:
\[ [\lambda] = L \]

3. **Check the Unit of \( c \):**
- The speed \( c \) is measured in meters per second:
\[ [c] = LT^{-1} \]

4. **Examine the Argument of the Sine Function:**
- The argument of the sine function is \( \frac{2\pi}{\lambda}(ct - x) \).
- The term \( ct \) has units:
\[ [ct] = [c][t] = (LT^{-1})(T) = L \]
- The term \( x \) has units:
\[ [x] = L \]
- Therefore, \( ct - x \) has units:
\[ [ct - x] = L - L = L \]

5. **Determine the Units of \( \frac{2\pi}{\lambda} \):**
- The unit of \( \lambda \) is \( L \), so:
\[ \left[\frac{2\pi}{\lambda}\right] = L^{-1} \]

6. **Combine the Units:**
- The entire expression \( \frac{2\pi}{\lambda}(ct - x) \) must be dimensionless because it is the argument of the sine function. Thus:
\[ \left[\frac{2\pi}{\lambda}(ct - x)\right] = L^{-1} \cdot L = 1 \quad (\text{dimensionless}) \]

7. **Conclusion:**
- From the analysis, we can conclude that the units of \( y \), \( A \), and \( x \) are the same, which is meters. Therefore, the correct statement is:
- The unit of \( \lambda \) is the same as that of \( x \) and \( A \).

### Final Answer:
The true statement is: The unit of \( \lambda \) is the same as that of \( x \) and \( A \).