A ball with a mass of #2# #kg # and velocity of #5# # ms^-1# collides with a second ball with a mass of #7# #kg# and velocity of #- 4# #ms^-1#. If #40%# of the kinetic energy is lost, what are the final velocities of the balls?
Momentum is conserved, but in this instance kinetic energy is not... but we know how much is lost. The solution is
Momentum prior to impact:
Kinetic energy prior to impact:
Momentum following the impact:
kinetic energy following impact:
Replace in Equation 2:
This is a quadratic equation that can be resolved in your preferred way or by using the quadratic formula:
Equation 1's result after these values are substituted is:
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Using the principle of conservation of momentum and the given information, we can find the final velocities of the balls.
Let ( m_1 = 2 , \text{kg} ) be the mass of the first ball, ( v_1 = 5 , \text{m/s} ) be its initial velocity, ( m_2 = 7 , \text{kg} ) be the mass of the second ball, and ( v_2 = -4 , \text{m/s} ) be its initial velocity.
The total initial momentum is ( p_{\text{initial}} = m_1v_1 + m_2v_2 ).
Using the principle of conservation of momentum, the total final momentum is equal to the initial momentum: ( p_{\text{final}} = p_{\text{initial}} ).
We can calculate the final velocities of the balls using the given information and the fact that kinetic energy is lost.
Let ( v_{1f} ) and ( v_{2f} ) be the final velocities of the first and second balls, respectively.
We also know that 40% of the kinetic energy is lost, so the final kinetic energy is ( 0.6 \times \text{initial kinetic energy} ).
Given that kinetic energy is ( \frac{1}{2}mv^2 ), we can write the equation for kinetic energy before and after the collision.
For ball 1: [ \frac{1}{2}m_1v_1^2 = \frac{1}{2}m_1{v_{1f}}^2 ]
For ball 2: [ \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_2{v_{2f}}^2 ]
We can solve these equations to find ( 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.
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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- A ball with a mass of #5kg# moving at #4 m/s# hits a still ball with a mass of #7 kg#. If the first ball stops moving, how fast is the second ball moving?
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