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}mv^2.

Object A has mass $8$ kilograms and positive speed $v$ . Object B has mass $2$ kilograms and speed $v+3$ meters per second. The two objects have the same kinetic energy.

What is the value of $v$ ?

For kinetic energy, mass multiplies the square of speed, so equal energies don't mean equal speeds. Substitute each object's values into the formula, set the results equal, and use a Desmos regression restricted to positive speeds. To turn a mass ratio into a speed ratio, take its square root. The factor of one-half multiplies both energies; it doesn't halve either speed.

Hints

Hint 1
Substitute each object's mass and entire speed into the kinetic-energy formula. The square covers all of B's speed, including the added amount. Which two expressions must match?
Hint 2
Both sides of the energy equation contain a common factor of $\frac12$ . Cancel it from both whole sides, not from a speed. What one-variable equation is left?
Hint 3
A regression asks Desmos for a value that makes two expressions match. Replace $v$ with $v_1$ and $=$ with $\sim$ . Restrict $v_1>0$ because a negative speed isn't allowed.

Step-by-step

Approach 1: Set the energies equal and use Desmos
Step 1
Write the equal-energy equation
Put each object's mass and speed into the kinetic-energy formula. Since their energies are the same, set the results equal:
$\frac12(8)v^2=\frac12(2)(v+3)^2.$
Keep the parentheses: the square covers B's entire speed, $v+3$ .
Step 2
Cancel the shared factor
The $\frac12$ is a common factor on both sides. Cancel it from the whole equation:
$8v^2=2(v+3)^2.$
Step 3
Find the positive speed
Type $8v_1^2\sim2(v_1+3)^2\{v_1>0\}$ in Desmos. Replacing $v$ with $v_1$ and $=$ with $\sim$ makes this a regression: Desmos finds a value that makes the sides match. The braces require a positive speed; a negative value could also satisfy an equation with squares. Under PARAMETERS, Desmos shows $v_1=3$ . So Object A's speed $v$ is $3$ meters per second. Choice D.

Approach 2: Compare the speeds by hand
Step 1
Compare squared speeds
Start from the equal-energy equation $8v^2=2(v+3)^2$ . Divide both sides by $2$ :
$4v^2=(v+3)^2.$
Step 2
Use the positive speeds
Both $v$ and $v+3$ are positive speeds, so take the positive square root of each side:
$2v=v+3.$
The square root turns the factor of $4$ between squared speeds into a factor of $2$ between speeds.
Step 3
Solve for Object A's speed
Subtract $v$ from both sides:
$v=3.$
So Object A's speed is $3$ meters per second. Choice D.

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

Strategy

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Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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