A particle is moving along the curve `x=at^(2)+bt+c`. If `ac=b^(2)`, then particle would be moving with uniform

Options

A. rotation
B. velocity
C. acceleration
D. retardation

Explanation

To solve the problem step by step, we start with the given equation of motion for the particle:

### Step 1: Identify the position function
The position of the particle is given by the equation:
\[ x(t) = at^2 + bt + c \]

### Step 2: Differentiate to find velocity
To find the velocity of the particle, we differentiate the position function with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(at^2 + bt + c) \]
Using the power rule of differentiation, we get:
\[ v(t) = 2at + b \]

### Step 3: Differentiate to find acceleration
Next, we differentiate the velocity function to find the acceleration:
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(2at + b) \]
Again, applying the power rule, we have:
\[ a(t) = 2a \]

### Step 4: Analyze the condition \( ac = b^2 \)
We are given the condition \( ac = b^2 \). However, we need to analyze the implications of our findings regarding the acceleration:
- From our calculations, we found that the acceleration \( a(t) = 2a \) is a constant (it does not depend on \( t \)).
- This means that the particle is moving with constant acceleration.

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

### Final Answer
The particle would be moving with uniform **acceleration**.

---