Transfer of momentum problem?


by greswd
Tags: momentum, transfer
greswd
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#1
Nov6-12, 06:11 AM
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Here's an interesting puzzle:

The whole scenario takes place in one dimension of space.

Ball B is at rest. Ball A moves with momentum [tex]p_a[/tex]

Ball A makes a perfectly elastic head-on collision with Ball B. Ball B moves off with momentum [tex]p_b[/tex]


Prove that [tex]|\frac{p_b}{p_a}|<2[/tex]
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A.T.
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Nov6-12, 06:58 AM
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Quote Quote by greswd View Post
Here's an interesting puzzle:

The whole scenario takes place in one dimension of space.

Ball B is at rest. Ball A moves with momentum [tex]p_a[/tex]

Ball A makes a perfectly elastic head-on collision with Ball B. Ball B moves off with momentum [tex]p_b[/tex]


Prove that [tex]|\frac{p_b}{p_a}|<2[/tex]
Momentum conservation :
[tex]{p_a} = {p_a}' + {p_b}[/tex]
[tex]1- \frac{{p_a}'}{p_a} = \frac{p_b}{p_a}[/tex]


Energy conservation (Ball B had intially no KE, so speed of ball A cannot increase):
[tex]|\frac{{p_a}'}{p_a}| < 1[/tex]
haruspex
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Nov7-12, 12:04 AM
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Quote Quote by A.T. View Post
[tex]|\frac{{p_a}'}{p_a}| < 1[/tex]
All true, but I don't see how that gets you to the answer required.
Using energy conservation more completely:
[tex]\frac{{p_a}'^2}{m_a}+\frac{{p_b}^2}{m_b}=\frac{{p_a}^2}{m_a}[/tex]
Combining with momentum eqn. we get
[tex]\frac{p_b}{p_a}=\frac{2m_b}{m_a+m_b}<2[/tex]

A.T.
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Nov7-12, 08:09 AM
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Transfer of momentum problem?


Quote Quote by haruspex View Post
All true, but I don't see how that gets you to the answer required.
You have to combine the two of course:
[tex]1- \frac{{p_a}'}{p_a} = \frac{p_b}{p_a}[/tex]
[tex]|\frac{{p_a}'}{p_a}| < 1[/tex]
This leads to:
[tex]0 < \frac{p_b}{p_a} < 2[/tex]

Quote Quote by haruspex View Post
Using energy conservation more completely:
[tex]\frac{{p_a}'^2}{m_a}+\frac{{p_b}^2}{m_b}=\frac{{p_a}^2}{m_a}[/tex]
Combining with momentum eqn. we get
[tex]\frac{p_b}{p_a}=\frac{2m_b}{m_a+m_b}<2[/tex]
Yes, the clean & complete way is to derive the ratio as function of the masses. Mine was just showing that the ratio is < 2.


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