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
Solution:
Step 1
Reformulate the given equation: $r x = qV + \frac{1}{2}m r^2$ as $qV + \frac{1}{2}m r^2 = x$.
Step 2
Combine like terms:
Step 2.1: Merge $m$ with $\frac{1}{2}$ to get $qV + \frac{m}{2} r^2 = x$.
Step 2.2: Combine $\frac{m}{2}$ with $r^2$ to obtain $qV + \frac{m r^2}{2} = x$.
Step 3
Isolate the quadratic term by subtracting $qV$ from both sides: $\frac{m r^2}{2} = x - qV$.
Step 4
Eliminate the fraction by multiplying by 2: $m r^2 = 2(x - qV)$.
Step 5
Simplify the equation:
Step 5.1: Simplify the left side by canceling out the 2: $m r^2 = 2(x - qV)$.
Step 5.2: Distribute the 2 on the right side: $m r^2 = 2x - 2qV$.
Step 6
Divide by $m$ to solve for $r^2$:
Step 6.1: Divide each term by $m$: $r^2 = \frac{2x}{m} - \frac{2qV}{m}$.
Step 6.2: Simplify the right side: $r^2 = \frac{2x - 2qV}{m}$.
Step 7
Take the square root of both sides to solve for $r$: $r = \pm \sqrt{\frac{2x - 2qV}{m}}$.
Step 8
Simplify the radical expression:
Step 8.1: Factor out the common factor of 2: $r = \pm \sqrt{\frac{2(x - qV)}{m}}$.
Step 8.2: Rewrite the square root as a fraction: $r = \pm \frac{\sqrt{2(x - qV)}}{\sqrt{m}}$.
Step 8.3: Rationalize the denominator: $r = \pm \frac{\sqrt{2(x - qV)}\sqrt{m}}{m}$.
Step 9
Present the complete solution:
Step 9.1: Positive root: $r = \frac{\sqrt{2(x - qV)m}}{m}$.
Step 9.2: Negative root: $r = -\frac{\sqrt{2(x - qV)m}}{m}$.
Step 9.3: The complete solution includes both roots: $r = \pm \frac{\sqrt{2(x - qV)m}}{m}$.
Knowledge Notes:
The problem involves solving a quadratic equation in the form of $r x = qV + \frac{1}{2}mr^2$. 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.
