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------------------------------------------------ An 8.00 kg object moving at 4.00 m/s in the positive x-direction has a one-dimensional collision with a 2.00 kg object moving at 3.00 m/s in the opposite direction. The final velocity of the 8.00 kg object is 2.00 m/s in the positive x-direction. What is the total kinetic energy of the two-mass system after the collision?

The total kinetic energy of the two-mass system after the collision is calculated using individual kinetic energies and summing them up, which results in a total kinetic energy of 41.00 Joules.

Explanation:

To find the total kinetic energy after the collision, we first need to calculate the kinetic energy for each object separately and then sum them. Since we know the final velocity of the 8.00 kg object, we only need to find the final velocity of the 2.00 kg object in order to calculate its kinetic energy after the collision.

The principle of conservation of momentum before and after the collision gives us:

m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2'

Where m1 and m1' are the mass and final velocity of the 8.00 kg object, and m2 and v2' are the mass and final velocity of the 2.00 kg object.

Substituting the given values, we have:

8.00 kg * 4.00 m/s + 2.00 kg * (-3.00 m/s) = 8.00 kg * 2.00 m/s + 2.00 kg * v2'

32.00 kg*m/s - 6.00 kg*m/s = 16.00 kg*m/s + 2.00 kg * v2'

26.00 kg*m/s = 16.00 kg*m/s + 2.00 kg * v2'

10.00 kg*m/s = 2.00 kg * v2'

v2' = 5.00 m/s in the positive x direction

Now that we have the final velocities, we can calculate the kinetic energy (KE) using the formula:

KE = 1/2 * m * v^2

KE1 = 1/2 * 8.00 kg * (2.00 m/s)^2 = 16.00 J

KE2 = 1/2 * 2.00 kg * (5.00 m/s)^2 = 25.00 J

The total kinetic energy after the collision is KE1 + KE2:

Total KE = 16.00 J + 25.00 J = 41.00 J