In this question, we are given that a particle is subjected to a force that increases linearly with time. The particle starts from rest at t = 0 and moves along the x-axis. We need to describe the motion of the particle during the time interval in which the force is acting on it.


When a particle is subjected to a force that increases linearly with time, it means the force acting on the particle is changing with time at a constant rate. Let's analyze the motion of the particle in the given scenario.

1. Force and Acceleration:
- According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
- In this case, since the force is increasing linearly with time, the acceleration of the particle will also increase linearly with time.
- As a result, the particle will experience a constant acceleration throughout the time interval in which the force is acting on it.

2. Velocity and Displacement:
- Since the particle starts from rest at t = 0, its initial velocity is zero.
- As the particle experiences a constant acceleration, its velocity will increase linearly with time.
- The displacement of the particle can be calculated using the equation of motion: s = ut + (1/2)at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
- As time progresses, the displacement of the particle will increase gradually due to its increasing velocity.

3. Graphical Representation:
- If we plot a graph of force versus time, it will be a straight line with a positive slope, indicating a linear increase in force over time.
- Similarly, if we plot a graph of velocity versus time, it will also be a straight line with a positive slope, indicating a linear increase in velocity over time.
- The displacement versus time graph will be a curve, as the displacement is determined by the area under the velocity versus time graph.


In conclusion, when a particle is subjected to a force that increases linearly with time, it experiences a constant acceleration. As a result, its velocity and displacement also increase linearly with time. The motion of the particle can be represented graphically as a straight line for velocity versus time and force versus time, and a curved line for displacement versus time.

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