Conservation of Energy and ball problem

In summary, to find the speed of Ball 2 after doubling the mass of Ball 1, we can use the conservation of momentum and energy equations to determine the ratio of the masses of the two balls. Then, by plugging in the initial and final speeds of Ball 2, we can solve for the final speed after doubling the mass of Ball 1. It is important to note that the balls must have different masses for this solution to work.
  • #1
cdbowman42
14
0
1.Ball 1 with an initial speed of 14 m/s has a perfectly elastic collision with Ball 2 that is initially at rest. Afterward, the speed of Ball 2 is 21 m/s. What will be the speed of Ball 2 if the mass of Ball 1 is doubled.



2. conservation of momentum: m1(v1i)=m1(v1f)+m2(v2f)

conservation of energy:.5m1(v1i)2=.5m1(v1f)2+.5m2(v2f)2

v1f=v1i(m1-m2)/(m1+m2)

v2f=v1i(2m1)/(m1+m2)



3. I'm not sure where to even begin because it seems I would need to know the mass of either ball to find the solution.
 
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  • #2
The problem provides the initial speed of ball 1 and the final speed of ball 2, you should be able to use those numbers to find the ratio of the ball's masses m2/m1. Once you find this quantity, you have enough information to determine the final speed of ball 2 if m1 doubles.

Hint: Solve for a=m2/m1 by creating ratios:
[tex]
b = \frac{v_{2f}}{v_{1i}} = \frac{2m_1}{m_1+m_2} = \frac{2}{1+a}
[/tex]
where I'm using a and b to replace the ratio quantities. Now you have a very simple algebraic equation.
 
  • #3
p1+p2=p1'+p2'
p2 = m2v2 = 0 (v=0)
Elastic means that the balls don't stick so p1' = 0 (v=0, again).

so you're left with p1=p2'
m1v1 = m2'v2'

the m's are proportional so doubling m1 will double m2 which means ? (you answer)
 
Last edited:
  • #4
iRaid said:
p1+p2=p1'+p2'
p2 = m2v2 = 0 (v=0)
Elastic means that the balls don't stick so p1' = 0 (v=0, again).

so you're left with p1=p2'
m1v1 = m2'v2'

Elastic does mean that the balls don't stick, however your analysis presupposes that the masses are identical. If they are not identical, then p1' is not zero. In fact, just by looking at the speeds given, you should be able to determine that m1 is a bit more massive than m2 (before you double the mass of m1).
 
  • #5
But let's assume that Ball 2 has a mass of 1 kg. Using the equations above, we can solve for the initial speed of Ball 1:

v1i = (m2(v2f) - m1(v1f)) / (m1 - m2)

Plugging in the values given in the problem, we get:

v1i = (1 kg * 21 m/s - 2 kg * 14 m/s) / (2 kg - 1 kg)

v1i = 7 m/s

Now, if we double the mass of Ball 1 to 4 kg, the equations become:

v1f = (4 kg * v1i - 1 kg * 21 m/s) / (4 kg + 1 kg) = 3.2 m/s

v2f = (4 kg * v1i) / (4 kg + 1 kg) = 5.6 m/s

Therefore, the speed of Ball 2 would be 5.6 m/s if the mass of Ball 1 is doubled.

This problem demonstrates the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed. In this case, the kinetic energy of Ball 1 is transferred to Ball 2 during the collision, resulting in an increase in its speed. Doubling the mass of Ball 1 does not change the total energy of the system, so the final speed of Ball 2 remains the same. This is a fundamental concept in physics that is used to explain many phenomena in the natural world.
 

1. What is the law of Conservation of Energy?

The law of conservation of energy states that energy cannot be created or destroyed, but it can be transferred or transformed from one form to another. This means that the total amount of energy in a closed system remains constant.

2. How does the law of Conservation of Energy apply to a ball problem?

In a ball problem, the law of conservation of energy can be used to determine the initial and final energy states of the ball. This can then be used to calculate the ball's speed, height, and other factors.

3. What are the different forms of energy that are conserved in a ball problem?

In a ball problem, the different forms of energy that are conserved include kinetic energy (energy of motion), potential energy (energy stored in an object's position), and thermal energy (energy associated with temperature).

4. How does friction affect the conservation of energy in a ball problem?

Friction can cause some of the ball's energy to be converted into thermal energy, which is not conserved in the system. This means that the total amount of energy at the beginning of the problem may not be the same as the total amount of energy at the end.

5. Can the law of Conservation of Energy be applied to real-life situations?

Yes, the law of conservation of energy is a fundamental principle of physics and is applicable to all physical systems, including real-life situations. It is often used in engineering, design, and energy production to ensure efficient and sustainable use of resources.

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