A particle moves rectilinearly. Its displacement x at time t is given by `x^(2) = at^(2) + b` where a and b are constants. Its acceleration at time t is proportional to

Options

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

Explanation

To solve the problem, we start with the given equation for displacement:

\[ x^2 = at^2 + b \]

where \( a \) and \( b \) are constants.

### Step 1: Differentiate the displacement equation
To find the acceleration, we first need to find the velocity. We can find the velocity by differentiating the displacement \( x \) with respect to time \( t \).

Using implicit differentiation:
\[ \frac{d}{dt}(x^2) = \frac{d}{dt}(at^2 + b) \]

This gives us:
\[ 2x \frac{dx}{dt} = 2at \]

### Step 2: Solve for velocity
From the equation above, we can solve for the velocity \( v = \frac{dx}{dt} \):
\[ v = \frac{at}{x} \]

### Step 3: Differentiate velocity to find acceleration
Next, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{at}{x}\right) \]

Using the quotient rule:
\[ a = \frac{x\frac{d}{dt}(at) - at\frac{dx}{dt}}{x^2} \]

### Step 4: Substitute \( \frac{dx}{dt} \)
We already found \( \frac{dx}{dt} = v = \frac{at}{x} \). Substituting this into the acceleration equation:
\[ a = \frac{x(a) - at\left(\frac{at}{x}\right)}{x^2} \]
\[ = \frac{ax - \frac{a^2t^2}{x}}{x^2} \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ a = \frac{ax^2 - a^2t^2}{x^3} \]

### Step 6: Analyze the proportionality
The acceleration \( a \) can be expressed as:
\[ a \propto \frac{ax^2 - a^2t^2}{x^3} \]

This shows that the acceleration is proportional to \( t^2 \) because the term \( a^2t^2 \) is present in the numerator.

### Conclusion
Thus, the acceleration at time \( t \) is proportional to \( t^2 \).

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