# A ball with a mass of #3 # #kg# is rolling at #3# # ms^-1# and elastically collides with a resting ball with a mass of #3# #kg#. What are the post-collision velocities of the balls?

Intuition and experience tell us all the momentum and energy will transfer from one ball to the other, but we should calculate to check.

The detailed calculation below shows that, indeed, after the collision the velocity of the first ball is

Momentum before the collision:

Momentum after the collision:

Conservation of momentum tells us these two things are equal to each other, so:

Divide through by 3 to make it simpler:

That's one equation in two unknowns. We'll need another equation if we're going to be able to solve this.

Kinetic energy before the collision:

Kinetic energy after the collision:

Kinetic energy is not always conserved, but in this case we are told that collision is elastic, which means that kinetic energy is conserved. That means these two things - before and after - are also equal to each other, so:

Let's divide through by 3/2 to make it neater:

Substitute this into Equation 2 and solve:

Subtracting 9 from both sides and rearranging:

Solve this quadratic equation using the quadratic formula or another method and you will find the roots:

Fascinatingly, these two roots relate to the situation before and after the collision! Initially the first ball had a velocity of 3, but after the collision it had a velocity of 0.

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To find the post-collision velocities of the balls, you can use the conservation of momentum and kinetic energy. Since the collision is elastic, both momentum and kinetic energy are conserved. The equation for conservation of momentum is: m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

And for conservation of kinetic energy: 0.5 * m1 * v1_initial^2 + 0.5 * m2 * v2_initial^2 = 0.5 * m1 * v1_final^2 + 0.5 * m2 * v2_final^2

By solving these two equations simultaneously, you can find the final velocities of both balls.

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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.

- The velocity of an object with a mass of #5 kg# is given by #v(t)= 2 t^2 + 9 t #. What is the impulse applied to the object at #t= 7 #?
- A ball with a mass of # 5 kg# is rolling at #8 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 #3 kg# moving at #8 m/s# hits a still ball with a mass of #4 kg#. If the first ball stops moving, how fast is the second ball moving?
- In a collision, is the net impulse experienced by an object equal to its momentum change?
- A ball with a mass of # 2 kg# is rolling at #25 m/s# and elastically collides with a resting ball with a mass of # 2 kg#. What are the post-collision velocities of the balls?

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