Question

How do you solve for ’v’ in \[{V_e} = \dfrac{1}{2}m{v^2}\] ?

Answer 1

Hint: Here we have simple equation containing three variables (unknown values) in this we need to solve for ‘v’ in \[{V_e} = \dfrac{1}{2}m{v^2}\] . We can solve for ‘m’ in \[{V_e} = \dfrac{1}{2}m{v^2}\] . We can solve this by the transpose method. That is the common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity).

Complete step-by-step answer:
Given, \[{V_e} = \dfrac{1}{2}m{v^2}\]
We transpose ‘m’ to the left hand side of the equation by dividing ‘m’ on the left hand side of the equation.
 \[\dfrac{{{V_e}}}{m} = \dfrac{1}{2}{v^2}\]
Rearranging we the equation we have,
 \[\dfrac{1}{2}{v^2} = \dfrac{{{V_e}}}{m}\]
We transpose ‘2’ to the right hand side of the equation by multiplying ‘2’ on the right hand side of the equation.
 \[{v^2} = \dfrac{{{V_e}}}{m} \times 2\]
Taking square root on both side of the equation,
 \[ \Rightarrow v = \pm \sqrt {\dfrac{{{V_e}}}{m} \times 2} \]
If we know the values of \[{V_e}\] and ‘m’ we can find the value of ‘v’.
So, the correct answer is “ \[ v = \pm \sqrt {\dfrac{{{V_e}}}{m} \times 2} \]”.

Note: Suppose if they asked us to solve for ‘m’ in \[{V_e} = \dfrac{1}{2}m{v^2}\] .
We transpose \[{v^2}\] to the left hand side of the equation by dividing \[{v^2}\] on the left hand side of the equation.
 \[\dfrac{{{V_e}}}{{{v^2}}} = \dfrac{1}{2}m\]
Rearranging we have,
 \[\dfrac{1}{2}m = \dfrac{{{V_e}}}{{{v^2}}}\]
We transpose ‘2’ to the right hand side of the equation by multiplying ‘2’ on the right hand side of the equation
 \[ \Rightarrow m = \dfrac{{{V_e}}}{{{v^2}}} \times 2\]
If we want to transpose a positive number to the other side of the equation we subtract the same number on that side (vice versa). Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose.