# A ball with a mass of #4 kg # and velocity of #3 m/s# collides with a second ball with a mass of #2 kg# and velocity of #- 8 m/s#. If #10%# of the kinetic energy is lost, what are the final velocities of the balls?

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To find the final velocities of the balls after the collision, we can use the principle of conservation of momentum and the concept of kinetic energy loss.

Let's denote the initial velocities of the balls as ( v_{1i} ) and ( v_{2i} ) and the final velocities as ( v_{1f} ) and ( v_{2f} ) for the first and second balls, respectively.

Using the conservation of momentum equation:

[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} ]

Substituting the given values:

[ 4 \times 3 + 2 \times (-8) = 4 \times v_{1f} + 2 \times v_{2f} ]

[ 12 - 16 = 4 \times v_{1f} + 2 \times v_{2f} ]

[ -4 = 4 \times v_{1f} + 2 \times v_{2f} ]

Now, we'll use the equation for kinetic energy loss:

[ \text{Kinetic energy loss} = \text{Initial kinetic energy} - \text{Final kinetic energy} ]

[ 0.1 = \frac{1}{2} m_1 (v_{1i}^2 + v_{1f}^2) + \frac{1}{2} m_2 (v_{2i}^2 + v_{2f}^2) - \frac{1}{2} m_1 v_{1i}^2 - \frac{1}{2} m_2 v_{2i}^2 ]

[ 0.1 = \frac{1}{2} \times 4 \times (3^2 + v_{1f}^2) + \frac{1}{2} \times 2 \times ((-8)^2 + v_{2f}^2) - \frac{1}{2} \times 4 \times 3^2 - \frac{1}{2} \times 2 \times (-8)^2 ]

[ 0.1 = 6 + v_{1f}^2 - 16 + v_{2f}^2 - 18 - 32 ]

[ 0.1 = v_{1f}^2 + v_{2f}^2 - 60 ]

[ v_{1f}^2 + v_{2f}^2 = 60.1 ]

Now, we have two equations:

[ -4 = 4 \times v_{1f} + 2 \times v_{2f} ] [ v_{1f}^2 + v_{2f}^2 = 60.1 ]

Solving these equations simultaneously will give us the final velocities ( v_{1f} ) and ( v_{2f} ).

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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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