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

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SAT Nonlinear Equations & Systems Question 119 | Free SAT Prep | Aniko

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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 $m$ ?

For formula-manipulation questions, treat the given equation like any other algebra equation: identify the variable you need to isolate, then systematically apply inverse operations in reverse order (clear fractions, divide off coefficients, then undo powers with roots). Keep track of exponents—if the variable is squared, your final step should involve a square root—and use any context (like speed being nonnegative) to decide between plus/minus roots when needed.

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Undo the one-half

In the formula $K = \tfrac{1}{2}mv^2$ , $v^2$ is being multiplied by $\tfrac{1}{2}$ . What operation can you do to both sides of the equation to remove the $\tfrac{1}{2}$ ?

Get $v^2$ alone first

After you clear the $\tfrac{1}{2}$ , $v^2$ will still be multiplied by $m$ . How can you undo multiplication by $m$ to get an expression for $v^2$ in terms of $K$ and $m$ ?

Turn $v^2$ into $v$

Once you have an equation of the form $v^2 = \text{(expression in }K\text{ and }m)$ , what operation do you need to do to both sides to solve for $v$ ? Remember that speed is not negative.

Desmos Guide

Enter the original formula

Type the original equation into Desmos: K = (1/2)*m*v^2. Then pick simple numerical values for $K$ and $m$ (for example, add sliders or set $K=18$ and $m=2$ on separate lines).

Test each answer choice

For each option, type an equation for $v$ (for example, v = 2K/m, v = sqrt(2K/m), etc.) using the same $K$ and $m$ values, and look at whether the original equation $K = (1/2)*m*v^2$ is satisfied (both sides equal) for that expression. The choice that makes the original equation true for those values is the correct formula for $v$ .

Step-by-step Explanation

Understand what you need to solve for

You are given the formula for kinetic energy:

K = \frac{1}{2}mv^2

The question asks you to solve this equation for $v$ . That means you want to rewrite the equation so that $v$ is alone on one side and everything else (only $K$ and $m$ ) is on the other side.

Isolate $v^2$

Right now, $v^2$ is multiplied by $\tfrac{1}{2}$ and by $m$ .

Undo the factor of $\tfrac{1}{2}$ by multiplying both sides by $2$ :

2K = mv^2

Now undo the multiplication by $m$ by dividing both sides by $m$ :

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

Now you have $v^2$ written in terms of $K$ and $m$ .

Solve for $v$ by undoing the square

To get from $v^2$ to $v$ , take the square root of both sides of the equation:

v = \pm \sqrt{\frac{2K}{m}}

Because $v$ represents speed, which is nonnegative, we choose the positive square root. So the correct expression for $v$ in terms of $K$ and $m$ is

v = \sqrt{\frac{2K}{m}}

Among the choices, this matches option B.

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Step-by-step Explanation

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