A particle initially at rest on a frictionless horizontal surface, is acted upon by a horizontal force which is constant is size and direction. A graph is plotted between the work done (W) on the particle, against the speed of the particle, (v). If there are no other horizontal forces acting on the particle the graph would look like

Options

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Explanation

To solve the problem, we need to analyze the relationship between the work done (W) on a particle and its speed (v) when a constant horizontal force is applied to it.

### Step-by-Step Solution:

1. **Understanding Work Done**:
The work done (W) on an object is defined as the change in kinetic energy (KE) of that object. Mathematically, this is expressed as:
\[ W = \Delta KE = KE_{final} - KE_{initial} \]
Since the particle starts from rest, its initial kinetic energy is zero.

2. **Kinetic Energy Formula**:
The kinetic energy (KE) of an object with mass \( m \) moving at speed \( v \) is given by:
\[ KE = \frac{1}{2} mv^2 \]

3. **Calculating Work Done**:
Since the initial kinetic energy is zero, the work done on the particle when it reaches speed \( v \) is:
\[ W = KE_{final} - KE_{initial} = \frac{1}{2} mv^2 - 0 = \frac{1}{2} mv^2 \]

4. **Relating Work Done to Speed**:
From the equation \( W = \frac{1}{2} mv^2 \), we can see that work done is directly proportional to the square of the speed:
\[ W \propto v^2 \]

5. **Graphing the Relationship**:
The relationship \( W \propto v^2 \) indicates that if we plot work done (W) on the y-axis against speed (v) on the x-axis, the graph will be a parabola opening upwards. This is because the equation can be rewritten as:
\[ W = \frac{1}{2} m v^2 \]
which is a quadratic equation in terms of \( v \).

6. **Conclusion**:
Therefore, the graph of work done (W) versus speed (v) will be a parabolic curve.

### Final Answer:
The graph of work done (W) against the speed of the particle (v) will look like a parabola.