Question 119·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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$K = \dfrac{1}{2}mv^2$

The kinetic energy $K$ , in joules, of an object is modeled by the formula above, where $m$ is the mass of the object, in kilograms, and $v$ is its speed, in meters per second. Which of the following expresses $v$ in terms of $K$ and $m$ ?

The phrase “express $v$ in terms of $K$ and $m$ ” signals a nonlinear formula rearrangement: keep $K$ and $m$ as letters and get $v$ alone. Clear the factors multiplying $v^2$ , then undo the square. A Desmos sample can help distinguish $v$ from $v^2$ , but it cannot prove a formula for every object. The main trap is stopping at $v^2$ .

Hints

Hint 1
Keep $K$ and $m$ as letters while you isolate $v$ . In the given formula, $v^2$ is multiplied by both $m$ and $\frac12$ . What operations will clear those factors?
Hint 2
Once you have $v^2$ alone, you still need $v$ . A square root undoes a square, but squaring hides whether the original value was positive or negative.
Hint 3
The problem calls $v$ a speed. Speeds cannot be negative, so which sign should you keep after taking the square root?

Step-by-step

Isolate the square, then the speed
Step 1
Clear the factor of one-half
The $\frac12$ multiplies $mv^2$ , so multiply both sides by $2$ :
$2K=mv^2.$
Step 2
Get the square alone
For an object with nonzero mass, divide both sides by $m$ to isolate $v^2$ :
$\frac{2K}{m}=v^2.$
Step 3
Undo the square
Take square roots of both sides. A positive and a negative number have the same square, so keep both signs for now:
$v=\pm\sqrt{\frac{2K}{m}}.$
Step 4
Check what the isolated expression means
Choose $m=4$ and $K=8$ as a check, not as values supplied by the question. Type $m=4$ , $K=8$ , $K\sim\frac12mv_1^2\{v_1>0\}$ , and $\frac{2K}{m}$ . The subscript and $\sim$ tell Desmos to solve for speed, while the restriction selects a positive value. Under PARAMETERS, it shows $v_1=2$ ; the last line prints $4$ . So $\frac{2K}{m}$ is the squared speed, not the speed.
Step 5
Choose the sign that fits a speed
Speed is nonnegative, so choose the positive square-root branch:
$v=\sqrt{\frac{2K}{m}}.$
Isolating $v^2$ is not the same as isolating $v$ . Choice B.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 119·Medium·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