# A ball with a mass of #4 kg# moving at #6 m/s# hits a still ball with a mass of #9 kg#. If the first ball stops moving, how fast is the second ball moving? How much kinetic energy was lost as heat in the collision?

We can answer this question by recognizing that momentum is conserved, but kinetic energy is only conserved in a fully elastic collision.

The 4 kg ball has all of the momentum prior to the collision:

Since momentum is conserved, all of the momentum is transferred to the second ball when the first ball stops:

In case it is required, the complete equation is:

Since energy is required in the second part of the equation, we must determine the difference between the two kinetic energies:

Thus, in the collision, 40 J of kinetic energy is lost to entropy.

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To find the final velocity of the second ball, you can use the principle of conservation of momentum. The total momentum before the collision equals the total momentum after the collision.

The equation for conservation of momentum is: m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

Substitute the given values: (4 kg * 6 m/s) + (9 kg * 0 m/s) = (4 kg * 0 m/s) + (9 kg * v2_final)

Solve for v2_final:
24 kg*m/s = 9 kg * v2_final
v2_final = 24 kg*m/s / 9 kg
v2_final = 2.67 m/s

To find the amount of kinetic energy lost as heat in the collision, you need to calculate the initial and final kinetic energies and find the difference.

Initial kinetic energy: KE_initial = (1/2) * m1 * v1_initial^2 + (1/2) * m2 * v2_initial^2 KE_initial = (1/2) * 4 kg * (6 m/s)^2 + (1/2) * 9 kg * (0 m/s)^2

Final kinetic energy: KE_final = (1/2) * m1 * v1_final^2 + (1/2) * m2 * v2_final^2 KE_final = (1/2) * 4 kg * (0 m/s)^2 + (1/2) * 9 kg * (2.67 m/s)^2

Calculate the difference to find the kinetic energy lost as heat: KE_lost = KE_initial - KE_final

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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