A moving mass of $8 k g$ collides elastically with a stationary mass of $2 k g$ .If KE be the initial kinetic energy of the moving mass, what will be the kinetic energy left with it after the collision?

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Solution

Step 1: Given
Mass of the first object: $m_{1} = 8 k g$
Mass of the second object: $m_{2} = 2 k g$
The initial velocity of the first object is $u_{1}$ .
The initial velocity of the second object: $u_{2} = 0$ (since, second object is at rest)
The final velocity of the first object is $v_{1}$ .
The final velocity of the second object is $v_{2}$
The kinetic energy of the collision is $K E$
Step 2: Formula Used
$K i n e t i c E n e r g y = \frac{1}{2} m v^{2}$
In an elastic collision,
The linear momentum is conserved, so $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ .
The kinetic energy is conserved, so $\frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2} = \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2}$ .
On solving these equations: $v_{1} = \frac{m_{1} - m_{2}}{(m_{1} + m_{2})} u_{1} + \frac{2 m_{2}}{(m_{1} + m_{2})} u_{2}$
Step 3: Find the initial kinetic energy of mass 8 kg.
Calculate the initial velocity of mass $8 k g$ in terms of KE
$K i n e t i c E n e r g y = \frac{1}{2} m v^{2} \Rightarrow K E = \frac{1}{2} \times 8 \times {u_{1}}^{2} \Rightarrow K E = 4 {u_{1}}^{2} \Rightarrow u_{1} = \frac{\sqrt{K E}}{2}$
Calculate the final velocity of mass $8 k g$ using the formula
$v_{1} = \frac{m_{1} - m_{2}}{(m_{1} + m_{2})} u_{1} + \frac{2 m_{2}}{(m_{1} + m_{2})} u_{2} = \frac{8 - 2}{(8 + 2)} \times \frac{\sqrt{K E}}{2} + \frac{2 \times 2}{(8 + 2)} \times 0 = \frac{6}{10} \frac{\sqrt{K E}}{2} + 0 = 0.3 \sqrt{K E}$
Calculate the final kinetic energy of mass $8 k g$ .
$K i n e t i c E n e r g y = \frac{1}{2} m {v_{1}}^{2} = \frac{1}{2} \times 8 \times {(0.3 \sqrt{K E})}^{2} = 0.36 K E$
Hence, the kinetic energy after collision is $0.36 K E$ .

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A moving mass of 8kg collides elastically with a stationary mass of 2kg.If KE be the initial kinetic energy of the moving mass, what will be the kinetic energy left with it after the collision?

A moving mass of $8 k g$ collides elastically with a stationary mass of $2 k g$ .If KE be the initial kinetic energy of the moving mass, what will be the kinetic energy left with it after the collision?