The kinetic energy (in joules) of an object with mass (in kilograms) and speed (in meters per second) is given by Which expression gives in terms of and ?

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SAT Nonlinear Equations & Systems Question 134 | Free SAT Prep | Aniko

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Question 134·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The kinetic energy $K$ (in joules) of an object with mass $m$ (in kilograms) and speed $v$ (in meters per second) is given by

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

Which expression gives $v$ in terms of $K$ and $m$ ?

When a question says "express [variable] in terms of" other variables, rewrite the given formula by isolating that variable step by step: undo constants and coefficients using inverse operations (add/subtract, multiply/divide), keep all other variables together like normal numbers, and be especially careful with exponents—if the variable is squared, you will need to solve for the squared form first and then take a square root, remembering to choose the physically meaningful sign when there is a context like speed.

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Focus on the variable you need

You need an expression for $v$ . Treat $K$ and $m$ as constants and think: what algebraic steps will get $v$ alone?

Undo the coefficient in front of $v^2$

Right now $v^2$ is multiplied by $\tfrac{1}{2}m$ . What operation will remove the $\tfrac{1}{2}$ so that $m v^2$ is by itself?

Get $v^2$ alone, then take a square root

Once you have an equation of the form $v^2 = \text{(expression with K and m)}$ , what operation do you apply to both sides to solve for $v$ ?

Desmos Guide

Assign sample values to K and m

In Desmos, pick simple positive values for $K$ and $m$ , such as typing K = 10 and m = 5. These will stand in for 'some kinetic energy' and 'some mass' while you test the answer choices.

Test each answer choice by recomputing K

For choice A, define v_A = 2*K*m. Then define K_A = 0.5*m*(v_A)^2. Compare K_A to your original K. If they are not equal, A is wrong. Repeat this process with v_B, v_C, and v_D using the expressions from choices B, C, and D, each time computing K_choice = 0.5*m*(v_choice)^2 and comparing it to K.

Identify which expression works

The correct formula for $v$ will make the recomputed kinetic energy K_choice exactly match the original K (for your chosen values of $K$ and $m$ , and it will continue to match if you change those values). The option whose K_choice consistently equals K is the correct answer.

Step-by-step Explanation

Identify the goal and the given equation

We are given the formula for kinetic energy:

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

We are asked to solve for $v$ in terms of $K$ and $m$ . That means we want to rewrite the equation so that $v$ is alone on one side and everything else is on the other side.

Isolate $v^2$

First, get rid of the factor $\tfrac{1}{2}$ .

Multiply both sides by $2$ :

2K = m v^2

Now divide both sides by $m$ to solve for $v^2$ :

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

So we have expressed $v^2$ in terms of $K$ and $m$ .

Solve for $v$

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

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

Because $v$ is a speed, it must be nonnegative, so we take the positive root:

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

This matches answer choice D.

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Step-by-step Explanation

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