To determine the change in velocity required to maintain the same kinetic energy when the mass of a body is increased to 4 times its original mass, we need to follow a step-by-step approach:
1. Understand the formula for Kinetic Energy (KE):
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
where [tex]\(m\)[/tex] is the mass and [tex]\(v\)[/tex] is the velocity of the body.
2. Set up the initial conditions:
Let's assume the original mass is [tex]\(m_1\)[/tex] and the original velocity is [tex]\(v_1\)[/tex].
3. Determine the initial Kinetic Energy:
[tex]\[
KE_{\text{initial}} = \frac{1}{2} m_1 v_1^2
\][/tex]
4. Establish the conditions after the mass change:
The mass of the body increases to 4 times the original mass. So, the new mass [tex]\(m_2\)[/tex] is:
[tex]\[
m_2 = 4 m_1
\][/tex]
Let the new velocity be [tex]\(v_2\)[/tex].
5. Set the final Kinetic Energy to be equal to the initial Kinetic Energy (to maintain the same KE):
[tex]\[
KE_{\text{final}} = \frac{1}{2} m_2 v_2^2
\][/tex]
Since the kinetic energy remains the same,
[tex]\[
KE_{\text{final}} = KE_{\text{initial}}
\][/tex]
Therefore,
[tex]\[
\frac{1}{2} m_1 v_1^2 = \frac{1}{2} 4 m_1 v_2^2
\][/tex]
6. Solve for the new velocity [tex]\(v_2\)[/tex]:
First, we can cancel out the common factors:
[tex]\[
m_1 v_1^2 = 4 m_1 v_2^2
\][/tex]
Divide both sides by [tex]\(m_1\)[/tex]:
[tex]\[
v_1^2 = 4 v_2^2
\][/tex]
To isolate [tex]\(v_2\)[/tex], divide both sides by 4:
[tex]\[
v_2^2 = \frac{v_1^2}{4}
\][/tex]
Take the square root of both sides to solve for [tex]\(v_2\)[/tex]:
[tex]\[
v_2 = \frac{v_1}{2}
\][/tex]
So, the new velocity [tex]\(v_2\)[/tex] should be half of the original velocity [tex]\(v_1\)[/tex] to maintain the same kinetic energy when the mass of the body is increased to 4 times its original mass. Thus, if the initial velocity was [tex]\(v_1\)[/tex], the new velocity should be [tex]\(v_1 / 2\)[/tex].
