# A ball with a mass of #5 kg # and velocity of #6 m/s# collides with a second ball with a mass of #3 kg# and velocity of #- 2 m/s#. If #40%# 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 equation for kinetic energy.

Let v1 and v2 be the final velocities of the first and second balls, respectively.

Using the conservation of momentum:

m1 * v1 + m2 * v2 = m1 * u1 + m2 * u2

Where: m1 = mass of the first ball = 5 kg m2 = mass of the second ball = 3 kg u1 = initial velocity of the first ball = 6 m/s u2 = initial velocity of the second ball = -2 m/s

And using the equation for kinetic energy:

KE = 0.5 * m * v^2

Where KE is the kinetic energy, m is the mass, and v is the velocity.

Since 40% of the kinetic energy is lost, the final kinetic energy will be 60% of the initial kinetic energy.

So, for the first ball: Initial KE1 = 0.5 * 5 * (6)^2 = 90 J Final KE1 = 0.6 * 90 = 54 J

And for the second ball: Initial KE2 = 0.5 * 3 * (-2)^2 = 6 J Final KE2 = 0.6 * 6 = 3.6 J

Now, we can use the equation for kinetic energy to find the final velocities:

For the first ball: 54 = 0.5 * 5 * v1^2 v1^2 = 54 / (0.5 * 5) v1^2 = 21.6 v1 ≈ √21.6 ≈ 4.64 m/s

For the second ball: 3.6 = 0.5 * 3 * v2^2 v2^2 = 3.6 / (0.5 * 3) v2^2 = 2.4 v2 ≈ √2.4 ≈ 1.55 m/s

Therefore, the final velocities of the balls are approximately 4.64 m/s for the first ball and 1.55 m/s for the second ball.

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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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