The displacement x of a particle moving along x-axis at time t is given by `x^2 =2t^2 + 6t`. The velocity at any time t is

Options

A. `(4t + 6)/x`
B. `(2t - x)/x`
C. `4t + 6`
D. `(2t + 3)/x`

Answer

To find the velocity of a particle whose displacement \( x \) is given by the equation \( x^2 = 2t^2 + 6t \), we can follow these steps:

### Step 1: Differentiate the displacement equation
We start with the given equation:
\[ x^2 = 2t^2 + 6t \]
To find the velocity, we need to differentiate \( x \) with respect to \( t \).

### Step 2: Use implicit differentiation
Differentiating both sides with respect to \( t \):
\[ \frac{d}{dt}(x^2) = \frac{d}{dt}(2t^2 + 6t) \]
Using the chain rule on the left side:
\[ 2x \frac{dx}{dt} = 4t + 6 \]

### Step 3: Solve for \( \frac{dx}{dt} \)
Now, we can isolate \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = \frac{4t + 6}{2x} \]

### Step 4: Substitute \( x \) back into the equation
From the original equation, we can express \( x \) in terms of \( t \):
\[ x = \sqrt{2t^2 + 6t} \]
Thus, we can substitute \( x \) back into the equation for velocity:
\[ \frac{dx}{dt} = \frac{4t + 6}{2\sqrt{2t^2 + 6t}} \]

### Final Answer
The velocity \( v \) at any time \( t \) is:
\[ v = \frac{4t + 6}{2\sqrt{2t^2 + 6t}} = \frac{2t + 3}{\sqrt{2t^2 + 6t}} \]

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