A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of `45^@` with the X-axis. The two motions are given by
`x=x_0 sinomegat` and `s=s_0 sin omegat`.
find the amplitude of the resultant motion.

Explanation

To find the amplitude of the resultant motion of a particle subjected to two simple harmonic motions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Motions**:
The two simple harmonic motions are given as:
- \( x = x_0 \sin(\omega t) \) (along the X-axis)
- \( s = s_0 \sin(\omega t) \) (along a line making an angle of \( 45^\circ \) with the X-axis)

2. **Understand the Geometry of the Problem**:
The second motion can be resolved into its components along the X and Y axes. Since it makes an angle of \( 45^\circ \), the components are:
- \( s_x = s_0 \sin(\omega t) \cos(45^\circ) = \frac{s_0}{\sqrt{2}} \sin(\omega t) \)
- \( s_y = s_0 \sin(\omega t) \sin(45^\circ) = \frac{s_0}{\sqrt{2}} \sin(\omega t) \)

3. **Resultant Displacement**:
The resultant displacement \( R \) can be expressed as:
\[ R = \sqrt{x^2 + s_x^2 + s_y^2 + 2s_x x} \]
However, since \( s_y \) is not contributing to the X-axis motion, we can simplify this to:
\[ R = \sqrt{x^2 + s_x^2 + 2s_x x} \]

4. **Substituting the Values**:
Substitute \( x \) and \( s_x \):
\[ R = \sqrt{(x_0 \sin(\omega t))^2 + \left(\frac{s_0}{\sqrt{2}} \sin(\omega t)\right)^2 + 2 \left(\frac{s_0}{\sqrt{2}} \sin(\omega t)\right)(x_0 \sin(\omega t))} \]

5. **Factor Out Common Terms**:
Since \( \sin(\omega t) \) is common, we can factor it out:
\[ R = \sin(\omega t) \sqrt{x_0^2 + \left(\frac{s_0}{\sqrt{2}}\right)^2 + 2 \left(\frac{s_0}{\sqrt{2}}\right)x_0} \]

6. **Simplifying the Expression**:
Simplifying the expression inside the square root:
\[ R = \sin(\omega t) \sqrt{x_0^2 + \frac{s_0^2}{2} + \sqrt{2} s_0 x_0} \]

7. **Finding the Amplitude**:
The amplitude \( A \) of the resultant motion is the coefficient of \( \sin(\omega t) \):
\[ A = \sqrt{x_0^2 + \frac{s_0^2}{2} + \sqrt{2} s_0 x_0} \]

### Final Result:
The amplitude of the resultant motion is:
\[ A = \sqrt{x_0^2 + \frac{s_0^2}{2} + \sqrt{2} s_0 x_0} \]