Question 913·Easy·Math

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The kinetic energy of an object is given by

K = \tfrac12 m v^2,

where $K$ is the kinetic energy, $m$ is the mass of the object, and $v$ is the speed of the object.

Which of the following equations gives the speed $v$ in terms of $K$ and $m$ ?

The words in terms of signal rearranging a formula: get the requested letter alone while the other letters stay in the answer. Work backward through the operations on that letter. Undo the multiplier before undoing the square. If the letter represents a speed, keep its nonnegative value after taking the square root.

Hints

Hint 1
The coefficient, or factor multiplying $v^2$ , is $\frac12m$ . Remove that factor from both sides before you undo the square. What happens if you multiply both sides by $2$ first?
Hint 2
Once $v^2$ is alone, a square root undoes the square. Squaring hides whether a value was positive or negative. Since $v$ is a speed, which sign makes sense?

Step-by-step

Isolate the speed by hand and check with Desmos
Step 1
Find a test speed from the formula
Choose positive test values $K=8$ and $m=4$ to check the result later. Type $K=8$ , $m=4$ , and $K\sim\frac12(4)v_1^2\{v_1>0\}$ in Desmos. Use $4$ instead of $m$ in the regression so Desmos fits only $v_1$ ; the $m=4$ line will be used to check the result later. The $\sim$ tells Desmos to find a value of $v_1$ that fits the equation, and the braces keep this speed positive. Desmos reports $v_1=2$ under PARAMETERS.
Step 2
Remove the factor of one-half
In $K=\tfrac12mv^2$ , one-half multiplies $mv^2$ . Multiply both sides by $2$ :
$2K=mv^2.$
Step 3
Get the square alone
$m$ still multiplies $v^2$ . Divide both sides by $m$ , since an object's mass is nonzero:
$\frac{2K}{m}=v^2.$
Step 4
Undo the square
A square root undoes squaring. Squaring hides the sign, but speed can't be negative, so take the nonnegative root: $v=\sqrt{\frac{2K}{m}}$ . Type $\sqrt{\frac{2K}{m}}$ in Desmos. For the test values, it prints $2$ , matching the speed found from the given formula. This gives the object's speed in terms of its kinetic energy and mass. Choice B.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 913·Easy·Math

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