A ball with a mass of #3 kg# is rolling at #3 m/s# and elastically collides with a resting ball with a mass of #1 kg#. What are the post-collision velocities of the balls?
Equations of conservation of energy and momentum.
As wikipedia suggests:
[Equations' source]
Derivation
Conservation of momentum and energy state:
Momentum
Energy
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To find the post-collision velocities of the balls, we can use the principle of conservation of momentum and kinetic energy. The equation for the conservation of momentum is:
(m1v1_initial + m2v2_initial = m1v1_final + m2v2_final)
Using the given values:
(3 kg * 3 m/s + 1 kg * 0 m/s = 3 kg * v1_final + 1 kg * v2_final)
Solving for v1_final and v2_final, we find:
(9 kg*m/s = 3 kg * v1_final + 1 kg * v2_final)
Since the collision is elastic, kinetic energy is conserved as well:
(0.5 * m1 * v1_initial^2 + 0.5 * m2 * v2_initial^2 = 0.5 * m1 * v1_final^2 + 0.5 * m2 * v2_final^2)
Plugging in the given values:
(0.5 * 3 kg * (3 m/s)^2 + 0.5 * 1 kg * (0 m/s)^2 = 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)
Solving for v1_final and v2_final, we find:
(13.5 J = 4.5 J + 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)
(9 J = 0.5 * 3 kg * v1_final^2 + 0.5 * 1 kg * v2_final^2)
Using the momentum equation, we found (v1_final = 0 m/s) and (v2_final = 9 m/s).
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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.
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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