The displacement-time graph of a particle acted upon by a constant force is

Options

A. a straight line
B. a circle
C. a parabola
D. any curve depending upon initial conditions

Explanation

To solve the question regarding the displacement-time graph of a particle acted upon by a constant force, we can follow these steps:

### Step 1: Understand the relationship between force, mass, 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. This can be expressed as:
\[ F = ma \]

### Step 2: Relate acceleration to displacement and time
When a constant force acts on a particle, it causes a constant acceleration. The equations of motion describe the relationship between displacement (s), initial velocity (u), time (t), and acceleration (a). One of the key equations is:
\[ s = ut + \frac{1}{2} a t^2 \]

### Step 3: Analyze the equation for displacement
In the equation \( s = ut + \frac{1}{2} a t^2 \), we can see that:
- The term \( ut \) represents the linear displacement due to the initial velocity.
- The term \( \frac{1}{2} a t^2 \) represents the additional displacement due to constant acceleration.

### Step 4: Identify the shape of the graph
The equation \( s = ut + \frac{1}{2} a t^2 \) is a quadratic equation in terms of time \( t \). When plotted on a graph with displacement (s) on the y-axis and time (t) on the x-axis, this equation will produce a parabolic curve.

### Step 5: Conclusion
Since the displacement-time graph of a particle acted upon by a constant force is a quadratic function of time, we conclude that the graph is a parabola. Therefore, the correct answer is:
**Option C: Parabola.**

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