Correct Answer - Option 3 : 2(√2 + 1)m\s
CONCEPT:
Kinetic energy
- The energy possessed by a body due to the virtue of its motion is called kinetic energy.
\(\Rightarrow KE=\frac{1}{2}mv^{2}\)
Where KE = kinetic energy, m = mass and v = velocity
CALCULATION:
Given Δv = v2 - v1 = 2m/s and KE2 = 2KE1
Where Δv = increase in speed
\(\Rightarrow KE_{1}=\frac{1}{2}mv_{1}^{2}\) -----(1)
\(\Rightarrow KE_{2}=\frac{1}{2}mv_{2}^{2}\) -----(2)
By equation 1 and equation 2,
\(\Rightarrow KE_{2}=2KE_{1}\)
\(\Rightarrow \frac{1}{2}mv_{2}^{2}=2\times\frac{1}{2}mv_{1}^{2}\)
\(\Rightarrow v_{2}^{2}=2v_{1}^{2}\)
\(\Rightarrow v_{2}=\pm\sqrt{2}v_{1}\) -----(3)
Since,
\(\Rightarrow \Delta v = v_{2}-v_{1}=2m/s\)
By equation 3,
\(\Rightarrow \sqrt{2}v_{1}-v_{1}=2m/s\)
\(\Rightarrow v_{1}=\frac{2}{\sqrt{2}-1}\) -----(4)
By dividing and multiplying \(\sqrt{2}+1\) in equation 4,
\(\Rightarrow v_{1}=\frac{2}{\sqrt{2}-1}\times\frac{\sqrt{2}+1}{\sqrt{2}+1}\)
\(\Rightarrow v_{1}=\frac{2(\sqrt{2}+1)}{\sqrt{2}^2-1^2}\)
\(\Rightarrow v_{1}=2(\sqrt{2}+1)\)m/s
- Hence, option 3 is correct.