If the distance covered by a particle is given by the relation `x=at^(2)`. The particle is moving with : (where a is constant)

Options

A. constant acceleration
B. zero acceleration
C. variable acceleration
D. none of these

Answer

To determine the type of motion of a particle whose distance covered is given by the equation \( x = at^2 \), where \( a \) is a constant, we can follow these steps:

### Step 1: Understand the given equation
The equation \( x = at^2 \) represents the displacement of the particle as a function of time \( t \). Here, \( a \) is a constant.

### Step 2: Find the velocity
To find the velocity of the particle, we differentiate the displacement \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d(at^2)}{dt} \]
Using the power rule of differentiation, we get:
\[ v = a \cdot \frac{d(t^2)}{dt} = a \cdot 2t = 2at \]

### Step 3: Find the acceleration
Next, we find the acceleration by differentiating the velocity \( v \) with respect to time \( t \):
\[ a = \frac{dv}{dt} = \frac{d(2at)}{dt} \]
Again applying the differentiation:
\[ a = 2a \cdot \frac{d(t)}{dt} = 2a \cdot 1 = 2a \]

### Step 4: Analyze the acceleration
Since \( a \) is a constant, \( 2a \) is also a constant. This indicates that the acceleration of the particle does not change over time.

### Conclusion
Since the acceleration is constant, we conclude that the particle is moving with constant acceleration.

### Final Answer
The particle is moving with constant acceleration.

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