The velocity of a particle moving on the x-axis is given by `v=x^(2)+x`, where x is in m and v in m/s. What is its position (in m) when its acceleration is `30m//s^(2)`.

Answer

To solve the problem, we need to find the position \( x \) of a particle when its acceleration \( a \) is \( 30 \, \text{m/s}^2 \). Given the velocity function:

\[ v = x^2 + x \]

we can find the acceleration by differentiating the velocity with respect to time. Let's go through the solution step by step.

### Step 1: Write down the given velocity equation
The velocity of the particle is given by:

\[ v = x^2 + x \]

### Step 2: Differentiate the velocity with respect to time
To find the acceleration, we need to differentiate the velocity with respect to time. We can use the chain rule:

\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \]

### Step 3: Differentiate \( v \) with respect to \( x \)
Now, we differentiate \( v \) with respect to \( x \):

\[ \frac{dv}{dx} = 2x + 1 \]

### Step 4: Substitute into the acceleration formula
Now substituting back into the acceleration formula:

\[ a = (2x + 1) \cdot v \]

Since \( v = x^2 + x \), we can substitute \( v \) into the equation:

\[ a = (2x + 1)(x^2 + x) \]

### Step 5: Expand the expression for acceleration
Now we expand this expression:

\[ a = (2x + 1)(x^2 + x) = 2x^3 + 2x^2 + x^2 + x = 2x^3 + 3x^2 + x \]

### Step 6: Set the acceleration equal to \( 30 \, \text{m/s}^2 \)
Now we set the expression for acceleration equal to \( 30 \):

\[ 2x^3 + 3x^2 + x = 30 \]

### Step 7: Rearrange the equation
Rearranging gives us:

\[ 2x^3 + 3x^2 + x - 30 = 0 \]

### Step 8: Solve the cubic equation using trial and error
We can solve this cubic equation using trial and error. Let's try some integer values for \( x \):

- For \( x = 1 \):

\[ 2(1)^3 + 3(1)^2 + 1 - 30 = 2 + 3 + 1 - 30 = -24 \quad (\text{not a solution}) \]

- For \( x = 2 \):

\[ 2(2)^3 + 3(2)^2 + 2 - 30 = 2(8) + 3(4) + 2 - 30 = 16 + 12 + 2 - 30 = 0 \quad (\text{this is a solution}) \]

### Step 9: Conclusion
Thus, the position \( x \) when the acceleration is \( 30 \, \text{m/s}^2 \) is:

\[ \boxed{2 \, \text{m}} \]