Solve for r x=qV+1/2mr^2

The question presents an equation in which x is defined as a function of r with qV and 1/2mr^2 as terms on the right side of the equation. The variable r appears to be the unknown in the equation, and the objective is to rearrange the equation to solve for the variable r, making r the subject of the formula. The equation seems to be describing a physical quantity or a property that relates mass (m), velocity (V), and some constants (q and x) together, possibly within the context of physics or a similar quantitative field. The task is to perform algebraic manipulations to isolate r and express it in terms of the other known quantities (x, q, V, and m).

$x = q V + \frac{1}{2} m r^{2}$

Answer

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Solution:

Step 1

Reformulate the given equation: $r x = q V + \frac{1}{2} m r^{2}$ as $q V + \frac{1}{2} m r^{2} = x$ .

Step 2

Combine like terms:

Step 2.1: Merge $m$ with $\frac{1}{2}$ to get $q V + \frac{m}{2} r^{2} = x$ .
Step 2.2: Combine $\frac{m}{2}$ with $r^{2}$ to obtain $q V + \frac{m r^{2}}{2} = x$ .

Step 3

Isolate the quadratic term by subtracting $q V$ from both sides: $\frac{m r^{2}}{2} = x - q V$ .

Step 4

Eliminate the fraction by multiplying by 2: $m r^{2} = 2 (x - q V)$ .

Step 5

Simplify the equation:

Step 5.1: Simplify the left side by canceling out the 2: $m r^{2} = 2 (x - q V)$ .
Step 5.2: Distribute the 2 on the right side: $m r^{2} = 2 x - 2 q V$ .

Step 6

Divide by $m$ to solve for $r^{2}$ :

Step 6.1: Divide each term by $m$ : $r^{2} = \frac{2 x}{m} - \frac{2 q V}{m}$ .
Step 6.2: Simplify the right side: $r^{2} = \frac{2 x - 2 q V}{m}$ .

Step 7

Take the square root of both sides to solve for $r$ : $r = \pm \sqrt{\frac{2 x - 2 q V}{m}}$ .

Step 8

Simplify the radical expression:

Step 8.1: Factor out the common factor of 2: $r = \pm \sqrt{\frac{2 (x - q V)}{m}}$ .
Step 8.2: Rewrite the square root as a fraction: $r = \pm \frac{\sqrt{2 (x - q V)}}{\sqrt{m}}$ .
Step 8.3: Rationalize the denominator: $r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m}$ .

Step 9

Present the complete solution:

Step 9.1: Positive root: $r = \frac{\sqrt{2 (x - q V) m}}{m}$ .
Step 9.2: Negative root: $r = - \frac{\sqrt{2 (x - q V) m}}{m}$ .
Step 9.3: The complete solution includes both roots: $r = \pm \frac{\sqrt{2 (x - q V) m}}{m}$ .

Knowledge Notes:

The problem involves solving a quadratic equation in the form of $r x = q V + \frac{1}{2} m r^{2}$ . The steps taken include isolating the quadratic term, simplifying the equation, and taking the square root to solve for the variable $r$ . The process involves algebraic manipulation, including distributing, factoring, and rationalizing the denominator. The solution provides both the positive and negative roots, as taking the square root of a number yields two possible values. The knowledge of quadratic equations, operations with radicals, and algebraic expressions is essential in solving this problem.