Let's carefully analyze each step of the equation given by the student and identify where mistakes have been made. The original equation is:
[tex]\[ k = \frac{1}{2} m v^2 \][/tex]
Our goal is to solve for [tex]\( v \)[/tex].
### Step-by-Step Solution:
Step 1: Start with the original equation.
[tex]\[ k = \frac{1}{2} m v^2 \][/tex]
Step 2: Isolate [tex]\( v^2 \)[/tex] by removing the coefficients.
Multiply both sides by 2 to get rid of the fraction:
[tex]\[ 2k = m v^2 \][/tex]
Step 3: Solve for [tex]\( v^2 \)[/tex].
Divide both sides by [tex]\( m \)[/tex]:
[tex]\[ v^2 = \frac{2k}{m} \][/tex]
Step 4: Solve for [tex]\( v \)[/tex].
Take the square root of both sides:
[tex]\[ v = \sqrt{\frac{2k}{m}} \][/tex]
This step-by-step solution correctly isolates [tex]\( v \)[/tex] without introducing additional terms or incorrect operations.
### Correcting the Student's Solution:
Let's point out the errors the student made in their process and explain how to correct them:
Incorrect step-by-step solution provided by the student:
1. The student hasn't shown necessary mathematical operations correctly.
[tex]\[ k = \frac{1}{2} m v^2 \][/tex]
This step is correctly started with the original equation.
2. Using the division property of equality incorrectly:
[tex]\[ k + m = \left(\frac{1}{2} m v^2\right) + m \][/tex]
This step is incorrect. Adding [tex]\( m \)[/tex] to both sides of the equation doesn't help isolate [tex]\( v \)[/tex]. This step introduces an unnecessary term and complicates the equation.
3. Further incorrect operations:
[tex]\[ \left(\frac{k}{2}\right) - 2 = \left(\frac{1}{2} v^2 \right) - 2 \][/tex]
Again, this step incorrectly modifies the original equation by dividing [tex]\( k \)[/tex] by 2 and subtracting 2 from both terms, which is not required to solve for [tex]\( v \)[/tex].
4. Incorrect application of the square root:
[tex]\[ \pm \frac{2 \sqrt{k}}{m} = \sqrt{v^2} \][/tex]
This step is incorrect. The square root should be taken directly from the simplified form of [tex]\( v^2 \)[/tex].
5. Incorrect simplification:
[tex]\[ \pm \frac{2 \sqrt{k}}{m} = v \][/tex]
This is an incorrect final form. The simplified version of the correct root for [tex]\( v \)[/tex] does not match the required manipulation of the original equation.
### Correct Statement:
To correct the process:
1. Start with the original equation.
[tex]\[ k = \frac{1}{2} m v^2 \][/tex]
2. Remove coefficients by appropriate multiplication and division:
[tex]\[ 2k = m v^2 \][/tex]
[tex]\[ v^2 = \frac{2k}{m} \][/tex]
3. Take the square root correctly:
[tex]\[ v = \sqrt{\frac{2k}{m}} \][/tex]
This approach correctly isolates [tex]\( v \)[/tex] using proper algebraic steps.
