The kinetic energy of an object is given by where is the kinetic energy, is the mass of the object, and is the speed of the object. Which of the following equations gives the speed in terms of and ?

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SAT Nonlinear Equations & Systems Question 4 | Free SAT Prep | Aniko

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Question 4·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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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 $m$ ?

When a physics-style formula is given and the question asks for one variable in terms of the others, treat it as a pure algebra problem: keep the symbols, systematically undo operations around the target variable in reverse order (divide to undo multiplication, subtract to undo addition, take roots to undo powers), and only at the end consider sign conventions (for quantities like speed or length, choose the positive root). This approach is faster and less error-prone than plugging in numbers or trying to manipulate all answer choices directly.

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Identify what you are solving for

You are given a formula for $K$ . Think about how you can rearrange it so that $v$ (or $v^2$ ) is alone on one side of the equation.

Undo the operations around $v^2$

In $K = \tfrac12 m v^2$ , $v^2$ is being multiplied by $\tfrac12 m$ . What operation would you apply to both sides to get $v^2$ by itself?

Handle the exponent on $v$

Once you have an equation of the form $v^2 =$ an expression in $K$ and $m$ , what operation will get you $v$ ? Also consider whether $v$ should be positive, negative, or either.

Desmos Guide

Set specific values for $K$ and $m$

In Desmos, enter two numbers such as m = 4 and K = 18. These act as constants so you can test each answer choice numerically.

Define each answer choice as a separate expression for v

Enter:

vA = 2K/m
vB = sqrt(2K/m)
vC = K*m/2
vD = sqrt(m/(2K)) These represent the four possible formulas for $v$ .

Test which candidate satisfies the original formula

For each candidate, type the right side of the kinetic energy formula using that $v$ :

R_A = 0.5*m*(vA)^2
R_B = 0.5*m*(vB)^2
R_C = 0.5*m*(vC)^2
R_D = 0.5*m*(vD)^2 Compare each $R_\text{letter}$ to the value of K. The one where $R_\text{letter}$ equals K (or matches it exactly) corresponds to the correct equation for v.

Step-by-step Explanation

Write the given formula and identify the goal

You are given the kinetic energy formula:

K = \tfrac12 m v^2

You need to solve this equation for $v$ , expressing $v$ in terms of $K$ and $m$ .

Isolate $v^2$ using inverse operations

Treat $K$ and $m$ as constants and undo the operations around $v^2$ .

Starting from

K = \tfrac12 m v^2,

first multiply both sides by $2$ :

2K = m v^2.

Then divide both sides by $m$ :

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

Now $v^2$ is isolated on one side.

Take the square root and choose the physically meaningful solution

From

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

solve for $v$ by taking the square root of both sides:

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

However, $v$ represents speed, which is always nonnegative, so we take only the positive root. The correct expression is

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

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation