To solve for the velocity [tex]\( v \)[/tex] when the kinetic energy [tex]\( KE \)[/tex] and mass [tex]\( m \)[/tex] are given, we can start by using the basic relation for kinetic energy:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
We need to isolate [tex]\( v \)[/tex] in this equation. Here are the steps in detail:
1. Start with the kinetic energy formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
2. To isolate [tex]\( v^2 \)[/tex], we first multiply both sides by 2 to get rid of the fraction:
[tex]\[ 2 KE = m v^2 \][/tex]
3. Next, divide both sides by [tex]\( m \)[/tex] to solve for [tex]\( v^2 \)[/tex]:
[tex]\[ \frac{2 KE}{m} = v^2 \][/tex]
4. Finally, to solve for [tex]\( v \)[/tex], take the square root of both sides:
[tex]\[ v = \sqrt{\frac{2 KE}{m}} \][/tex]
Thus, the correct formula to find the velocity [tex]\( v \)[/tex] when given the kinetic energy [tex]\( KE \)[/tex] and mass [tex]\( m \)[/tex] is:
[tex]\[ v = \sqrt{\frac{2 KE}{m}} \][/tex]
Among the given options:
1. [tex]\( v = \sqrt{\frac{1}{2}(K E)(m)} \)[/tex]
2. [tex]\( v = \sqrt{\frac{2 m}{K E}} \)[/tex]
3. [tex]\( v = \sqrt{K E(m)} \)[/tex]
4. [tex]\( v = \sqrt{\frac{2 K E}{m}} \)[/tex]
The correct option is:
[tex]\[ v = \sqrt{\frac{2 KE}{m}} \][/tex]
Therefore, the answer is:
[tex]\[ 4 \][/tex]
