A ball with a mass of #3 kg # and velocity of #9 m/s# collides with a second ball with a mass of #7 kg# and velocity of # 2 m/s#. If #80%# of the kinetic energy is lost, what are the final velocities of the balls?
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To solve for the final velocities of the balls after the collision, we can use the principle of conservation of momentum and the equation for kinetic energy.
Let ( v_1 ) be the final velocity of the first ball and ( v_2 ) be the final velocity of the second ball.
According to the conservation of momentum:
( m_1 \cdot v_1 + m_2 \cdot v_2 = m_1 \cdot u_1 + m_2 \cdot u_2 )
where:
 ( m_1 ) and ( m_2 ) are the masses of the first and second balls respectively,
 ( u_1 ) and ( u_2 ) are the initial velocities of the first and second balls respectively.
Given: ( m_1 = 3 ) kg, ( u_1 = 9 ) m/s, ( m_2 = 7 ) kg, ( u_2 = 2 ) m/s.
Then, ( v_1 ) and ( v_2 ) can be calculated.
After finding the velocities, we can use the equation for kinetic energy:
( KE = \frac{1}{2} m \cdot v^2 )
where:
 ( KE ) is the kinetic energy,
 ( m ) is the mass, and
 ( v ) is the velocity.
Given that 80% of the kinetic energy is lost:
( KE_{\text{final}} = 0.2 \times KE_{\text{initial}} )
After finding the final kinetic energies, we can solve for 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 fact that 80% of the kinetic energy is lost during the collision.

First, calculate the initial momentum of the system before the collision using the formula: [ \text{Initial momentum} = m_1 \cdot v_1 + m_2 \cdot v_2 ] where ( m_1 ) and ( m_2 ) are the masses of the first and second balls respectively, and ( v_1 ) and ( v_2 ) are their velocities.

Then, calculate the initial kinetic energy of the system before the collision using the formula: [ \text{Initial kinetic energy} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 ]

Since 80% of the kinetic energy is lost during the collision, the final kinetic energy will be 20% of the initial kinetic energy. Calculate the final kinetic energy.

Use the principle of conservation of momentum to find the final velocities of the balls after the collision: [ m_1 \cdot v_{1f} + m_2 \cdot v_{2f} = \text{Initial momentum} ]

Once you have the final velocities, you can check if the kinetic energy is 20% of the initial kinetic energy. If not, adjust the final velocities accordingly.
By following these steps, you can determine the final velocities of the balls after the collision.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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