Let's analyze the given steps and identify where the mistake might have occurred:
1. Start with the original equation:
[tex]\[
k = \frac{1}{2} mv^2
\][/tex]
This is the correct initial equation representing kinetic energy.
2. Use the division property of equality:
[tex]\[
t + m = \left(\frac{1}{2} mv^2\right) \div m
\][/tex]
This step doesn't make sense based on the initial equation. Division should be done directly on both sides without introducing [tex]\( t + m \)[/tex].
The correct division step would be:
[tex]\[
\frac{k}{m} = \frac{1}{2} v^2
\][/tex]
3. Use the multiplication property of equality:
[tex]\[
\left(\frac{k}{m}\right) \times 2 = \left(\frac{1}{2} v^2\right) \times 2
\][/tex]
This simplification correctly balances the equation:
[tex]\[
\frac{2k}{m} = v^2
\][/tex]
4. Use the square root property of equality:
[tex]\[
\pm \sqrt{\frac{2k}{m}} = \sqrt{v^2}
\][/tex]
This step is correct, as taking the square root of both sides to solve for [tex]\( v \)[/tex] gives:
[tex]\[
\pm \sqrt{\frac{2k}{m}} = v
\][/tex]
5. Simplify:
[tex]\[
\pm \sqrt{\frac{2k}{m}} = v
\][/tex]
This final simplification correctly identifies the velocity [tex]\( v \)[/tex]:
As per the analysis, It is evident from the steps above that the original steps were already correct except for step 2, where an unnecessary term was introduced. The correct approach should follow the proper balance of equations without additional arbitrary terms.
Therefore, the statement "The steps seem correct as given, no error in calculations shown." aligns with the accurate final simplification, given there is no actual mistake in the derived steps (except step 2's arbitrary term which was corrected). Each transformation, simplification, and property used adheres to mathematical standards.
