# A ball with a mass of #7 kg # and velocity of #4 m/s# collides with a second ball with a mass of #2 kg# and velocity of #- 6 m/s#. If #15%# of the kinetic energy is lost, what are the final velocities of the balls?

There may be a simpler way to solve this, but here's what I came up with:

Momentum is conserved in all collisions. In an inelastic collision, momentum is conserved as always, but energy is not; part of the kinetic energy is transformed into some other form of energy. Therefore, we have an inelastic collision.

The equation for momentum.

We can use momentum conversation and kinetic energy to find the final velocities of the balls.

Momentum conservation:

For multiple objects, we use superposition as with forces:

So we have:

We are given the following information:

We can begin by calculating the momentum before the collision:

Note that momentum is a vector quantity and the positive value indicates direction. Let's set to the right is the positive direction.

We can also calculate the initial kinetic energy before the collision:

This tells us that:

We now have two equations expressing final momentum and kinetic energy:

As the masses of the objects do not change, we can fill them in and simplify:

Simplifying:

Im going to multiply both sides by 7 to get rid of the fractions:

We have a quadratic equation which we can solve using the quadratic formula:

Therefore:

Therefore:

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The final velocities of the balls can be found using the principle of conservation of momentum and the conservation of kinetic energy. Let ( m_1 = 7 ) kg, ( v_{1i} = 4 ) m/s, ( m_2 = 2 ) kg, and ( v_{2i} = -6 ) m/s. Let ( v_{1f} ) and ( v_{2f} ) be the final velocities of the first and second balls, respectively.

According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:

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

Applying the conservation of kinetic energy, the initial kinetic energy minus the lost kinetic energy is equal to the final kinetic energy:

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

Since 15% of the kinetic energy is lost, the lost kinetic energy is ( 0.15 \times (\text{initial KE}) ):

[ \text{Lost KE} = 0.15 \times \left( \frac{1}{2} m_1 (v_{1i})^2 + \frac{1}{2} m_2 (v_{2i})^2 \right) ]

You can solve these equations to find 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.

- A ball with a mass of #6 kg# moving at #1 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?
- A ball with a mass of #2 kg# is rolling at #2 m/s# and elastically collides with a resting ball with a mass of #4 kg#. What are the post-collision velocities of the balls?
- A ball with a mass of #1 kg# moving at #9 m/s# hits a still ball with a mass of #6 kg#. If the first ball stops moving, how fast is the second ball moving? How much heat energy was lost in the collision?
- A ball with a mass of #6 kg# moving at #8 m/s# hits a still ball with a mass of #12 kg#. If the first ball stops moving, how fast is the second ball moving?
- An object with a mass of #4 kg# is traveling at #1 m/s#. If the object is accelerated by a force of #f(x) = x^2 -x +1 # over #x in [1, 9]#, where x is in meters, what is the impulse at #x = 2#?

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