The displacement x of a particle at time t moving along a straight line path is given by `x^(2) = at^(2) + 2bt + c` where a, b and c are constants. The acceleration of the particle varies as

Options

A. `x^(-1)`
B. `x^(-2)`
C. `x^(-3)`
D. `x^(-4)`

Explanation

To determine how the acceleration of a particle varies with time given the displacement equation \( x^2 = at^2 + 2bt + c \), we will follow these steps:

### Step 1: Differentiate the displacement equation
We start with the given equation:
\[ x^2 = at^2 + 2bt + c \]
To find the velocity, we differentiate both sides with respect to time \( t \):
\[ \frac{d}{dt}(x^2) = \frac{d}{dt}(at^2 + 2bt + c) \]
Using the chain rule on the left side, we have:
\[ 2x \frac{dx}{dt} = 2at + 2b \]
This simplifies to:
\[ x \frac{dx}{dt} = at + b \]
Thus, the velocity \( v \) is given by:
\[ v = \frac{dx}{dt} = \frac{at + b}{x} \]

### Step 2: Differentiate the velocity to find acceleration
Next, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ \frac{dv}{dt} = \frac{d}{dt}\left(\frac{at + b}{x}\right) \]
Using the quotient rule:
\[ \frac{dv}{dt} = \frac{x \cdot \frac{d}{dt}(at + b) - (at + b) \cdot \frac{dx}{dt}}{x^2} \]
Calculating \( \frac{d}{dt}(at + b) \) gives \( a \), and substituting \( \frac{dx}{dt} = v \):
\[ \frac{dv}{dt} = \frac{x \cdot a - (at + b) \cdot v}{x^2} \]

### Step 3: Substitute for \( v \)
Now, we substitute \( v = \frac{at + b}{x} \) into the acceleration equation:
\[ \frac{dv}{dt} = \frac{xa - (at + b) \cdot \frac{at + b}{x}}{x^2} \]
This simplifies to:
\[ \frac{dv}{dt} = \frac{xa - \frac{(at + b)^2}{x}}{x^2} \]
Multiplying through by \( x \) gives:
\[ \frac{dv}{dt} = \frac{ax^2 - (at + b)^2}{x^3} \]

### Step 4: Analyze the expression
The expression for acceleration can be rewritten as:
\[ a = \frac{ax^2 - (at + b)^2}{x^3} \]
This indicates that the acceleration is inversely proportional to \( x^3 \), as the numerator is a function of constants and \( x \).

### Conclusion
Thus, the acceleration of the particle varies as:
\[ a \propto \frac{1}{x^3} \]