Question 103·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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The kinetic energy of a moving object is given by

K = \frac{1}{2} m v^2,

where $K$ is the kinetic energy, $m$ is the mass, and $v$ is the speed. Which of the following expresses $v$ in terms of $K$ and $m$ , assuming $v > 0$ ?

To rearrange a squared formula, get the square of the requested variable alone, then take a square root. The condition that speed is positive tells you which root to keep. On a multiple-choice test, Desmos can also test the choices using convenient values. Don't stop when you've found an expression for squared speed instead of speed.

Hints

Hint 1
A formula for $v$ must work for every allowed mass and kinetic energy. Try $m=2$ so $\frac12m=1$ , and $K=9$ so the original equation becomes $v^2=9$ .
Hint 2
To test the original equation in Desmos, use $v_1$ for the unknown and $\sim$ instead of $=$ . Add $\{v_1>0\}$ because the question asks for a positive speed. What value does Desmos find?
Hint 3
Use that speed as a benchmark. Put your chosen $K$ and $m$ into the answer choices. Any choice that gives a different speed cannot express $v$ for every allowed pair of values.

Step-by-step

Approach 1: Test the choices with Desmos
Step 1
Find a speed to test against
A correct formula must work for any allowed values, so choose convenient values: $m=2$ makes $\frac12m=1$ , and let $K=9$ . Type those values and the given equation in Desmos, changing $v$ to $v_1$ and $=$ to $\sim$ . The regression reports $v_1=3$ under PARAMETERS. The restriction $v_1>0$ keeps the speed positive.
Step 2
Rule out expressions that give another speed
Type the first three expressions as a list, in the order shown. Desmos displays $[9,0.3333,2.25]$ . None equals $3$ , the speed found from the given formula, so none of those expressions works for these values.
Step 3
Check the remaining expression
Type the remaining expression. Desmos displays $3$ , so it matches the positive speed from the original equation. The other three choices have been ruled out, leaving $v=\sqrt{\frac{2K}{m}}$ . Choice D.

Approach 2: Rearrange the formula for every value
Step 1
Undo the one-half
In $K=\frac12mv^2$ , the squared speed is multiplied by $\frac12m$ . Undo the one-half first. Multiply both sides by $2$ :
$2K=mv^2.$
Step 2
Get squared speed alone
Mass is positive, so $m$ isn't zero. Divide both sides by $m$ :
$\frac{2K}{m}=v^2.$
This is $v^2$ , not yet the speed the question asks for.
Step 3
Choose the positive square root
A square can hide a sign: $3^2$ and $(-3)^2$ are both $9$ . Taking a square root could give either sign, but the given condition says $v>0$ . Take the positive square root:
$v=\sqrt{\frac{2K}{m}}.$
So this expresses the speed in terms of $K$ and $m$ . Choice D.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 103·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this