Answer:
The fraction of the total initial kinetic energy is lost during the collision is ![\dfrac{11}{17}\ J](https://tex.z-dn.net/?f=%5Cdfrac%7B11%7D%7B17%7D%5C%20J)
Explanation:
Given that,
Mass of one piece = 300 g
Speed of one piece = 1 m/s
Mass of other piece = 600 g
Speed of other piece = 0.75 m/s
We need to calculate the final velocity
Using conservation of energy
![m_{1}v_{1}+m_{2}v_{2}=(m_{1}+m_{2})v](https://tex.z-dn.net/?f=m_%7B1%7Dv_%7B1%7D%2Bm_%7B2%7Dv_%7B2%7D%3D%28m_%7B1%7D%2Bm_%7B2%7D%29v)
Put the value intro the formula
![300\times10^{-3}\times1+600\times10^{-3}\times(0.75)=(300\times10^{-3}+600\times10^{-3})v](https://tex.z-dn.net/?f=300%5Ctimes10%5E%7B-3%7D%5Ctimes1%2B600%5Ctimes10%5E%7B-3%7D%5Ctimes%280.75%29%3D%28300%5Ctimes10%5E%7B-3%7D%2B600%5Ctimes10%5E%7B-3%7D%29v)
![v=\dfrac{00\times10^{-3}\times1+600\times10^{-3}\times(-0.75)}{(300\times10^{-3}+600\times10^{-3})}](https://tex.z-dn.net/?f=v%3D%5Cdfrac%7B00%5Ctimes10%5E%7B-3%7D%5Ctimes1%2B600%5Ctimes10%5E%7B-3%7D%5Ctimes%28-0.75%29%7D%7B%28300%5Ctimes10%5E%7B-3%7D%2B600%5Ctimes10%5E%7B-3%7D%29%7D)
![v=-0.5\ m/s](https://tex.z-dn.net/?f=v%3D-0.5%5C%20m%2Fs)
We need to calculate the total initial kinetic energy
Using formula of kinetic energy
![K.E_{i}=\dfrac{1}{2}m_{1}v_{1}^2+\dfrac{1}{2}m_{2}v_{2}^2](https://tex.z-dn.net/?f=K.E_%7Bi%7D%3D%5Cdfrac%7B1%7D%7B2%7Dm_%7B1%7Dv_%7B1%7D%5E2%2B%5Cdfrac%7B1%7D%7B2%7Dm_%7B2%7Dv_%7B2%7D%5E2)
Put the value into the formula
![K.E_{i}=\dfrac{1}{2}\times300\times10^{-3}\times1^2+\dfrac{1}{2}\times600\times10^{-3}\times(0.75)^2](https://tex.z-dn.net/?f=K.E_%7Bi%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes300%5Ctimes10%5E%7B-3%7D%5Ctimes1%5E2%2B%5Cdfrac%7B1%7D%7B2%7D%5Ctimes600%5Ctimes10%5E%7B-3%7D%5Ctimes%280.75%29%5E2)
![K.E_{i}=0.31875\ J](https://tex.z-dn.net/?f=K.E_%7Bi%7D%3D0.31875%5C%20J)
We need to calculate the total final kinetic energy
Using formula of kinetic energy
![K.E_{f}=\dfrac{1}{2}(m_{1}+m_{2})v^2](https://tex.z-dn.net/?f=K.E_%7Bf%7D%3D%5Cdfrac%7B1%7D%7B2%7D%28m_%7B1%7D%2Bm_%7B2%7D%29v%5E2)
Put the value into the formula
![K.E_{f}=\dfrac{1}{2}\times(300\times10^{-3}+600\times10^{-3})\times(-0.5)^2](https://tex.z-dn.net/?f=K.E_%7Bf%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes%28300%5Ctimes10%5E%7B-3%7D%2B600%5Ctimes10%5E%7B-3%7D%29%5Ctimes%28-0.5%29%5E2)
![K.E_{f}=0.1125\ J](https://tex.z-dn.net/?f=K.E_%7Bf%7D%3D0.1125%5C%20J)
We need to calculate the energy lost during the collision
Using formula of energy lost
![energy\ lost=\dfrac{0.31875-0.1125}{0.31875}](https://tex.z-dn.net/?f=energy%5C%20lost%3D%5Cdfrac%7B0.31875-0.1125%7D%7B0.31875%7D)
![energy\ lost=\dfrac{11}{17}\ J](https://tex.z-dn.net/?f=energy%5C%20lost%3D%5Cdfrac%7B11%7D%7B17%7D%5C%20J)
Hence, The fraction of the total initial kinetic energy is lost during the collision is ![\dfrac{11}{17}\ J](https://tex.z-dn.net/?f=%5Cdfrac%7B11%7D%7B17%7D%5C%20J)