STATEMENT -1 `:`The force acting on a particle moving alongx -axis is `F = - alpha ( x+ vt)`, where `alpha` is a constant.
and
STATEMENT-2 `:` To an observer moving along x-axis with constant velocity v, it represents SHM.

Answer

To solve the problem, we need to analyze the two statements provided:

### Step 1: Understand the Force Equation
The force acting on the particle is given by:
\[ F = -\alpha (x + vt) \]
where \( \alpha \) is a constant, \( x \) is the displacement of the particle from the origin, and \( vt \) is a term that depends on time.

### Step 2: Identify the Observer's Frame of Reference
We have an observer moving along the x-axis with a constant velocity \( v \). We need to analyze how the force appears from this observer's frame.

### Step 3: Transform the Displacement
For the observer moving with velocity \( v \), the position of the particle can be expressed as:
\[ x_1 = vt \]
This means that the observer sees the particle's position relative to themselves as:
\[ x_{\text{relative}} = x - vt \]

### Step 4: Substitute into the Force Equation
Now, we substitute \( x_{\text{relative}} \) into the force equation:
\[ F = -\alpha (x_{\text{relative}} + vt) \]
Substituting \( x = x_{\text{relative}} + vt \):
\[ F = -\alpha ((x_{\text{relative}} + vt) + vt) \]
\[ F = -\alpha (x_{\text{relative}} + 2vt) \]

### Step 5: Analyze the New Force Equation
The new force equation becomes:
\[ F = -\alpha x_{\text{relative}} - 2\alpha vt \]
This shows that there is a time-dependent term \( -2\alpha vt \) present in the force equation.

### Step 6: Determine if it Represents SHM
For the system to represent Simple Harmonic Motion (SHM), the force must be proportional to the displacement and must not contain any explicit time dependence. The presence of the term \( -2\alpha vt \) indicates that the force is not purely dependent on the displacement \( x_{\text{relative}} \) alone.

### Conclusion
Since the force equation contains a term that depends on time, it does not represent SHM in the observer's frame of reference. Therefore, we conclude that:

- **Statement 1** is true (the force equation is correctly stated).
- **Statement 2** is false (it does not represent SHM for the observer).

### Final Answer
- **Statement 1**: True
- **Statement 2**: False

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