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Let's say that a $2\text{ kg}$ cart is rolling (frictionless) along a surface at $2\text{ ms}^{-1}$. A suspended $2\text{ kg}$ weight is then dropped onto it from above.

If we apply the law of conservation of momentum (in the horizontal axis) to figure out the velocity after the block is dropped onto the car, $$m_Cu_C+m_Wu_W=m_Cv+m_Wv$$ $$m_Cu_C+m_Wu_W=v(m_C+m_W)$$ $$\implies v=\frac{m_Cu_C+m_Wu_W}{m_C+m_W}$$ $$v=\frac{2 \times 2 + 2 \times 0}{2+2}$$ $$v=1\text{ ms}^{-1}$$

However if we apply the law of conservation of energy (between when the block is let go and after the landing),

$$E_{Cbefore}+E_{Wbefore}=E_{Cafter}+E_{Wafter}$$ $$E_{Cbefore}+0=E_{Cafter}+E_{Wafter}$$ $$\frac{1}{2}m_Cu_C^2=\frac{1}{2}m_Cv^2+\frac{1}{2}m_Wv^2$$ $$\frac{1}{2}\times 2\times 2^2=\frac{1}{2}\times 2\times v^2+\frac{1}{2}\times 2\times v^2$$ $$v^2=2$$ $$v\approx 1.41 \text{ ms}^{-1}$$

My questions are:

  • Where does this discrepancy come from?

  • If the answer something to do with the collision being an inelastic one, then why does the height that the weight is dropped from not play into the calculations?

nicoguaro
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Zack
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1 Answers1

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This is an inelastic collision, so kinetic energy is not conserved.

ultracoldgrl
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  • How do we know it is an inelastic collision? Where does the energy go? What if the question asked you to assume that the collision was elastic? – Zack May 27 '19 at 00:08
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    It's the definition! "A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision." – ultracoldgrl May 27 '19 at 00:09
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    The energy goes into other forms of energy other than kinetic energy such as binding energy/heat/etc. – ultracoldgrl May 27 '19 at 00:10
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    So you are saying that given the information that we have, there is no possible way for this collision to be elastic? – Zack May 27 '19 at 00:12
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    I like to think of elastic collisions as "bouncing" and inelastic as "sticking". Here's some more info: https://physics.stackexchange.com/questions/92051/how-can-momentum-but-not-energy-be-conserved-in-an-inelastic-collision/92391 – ultracoldgrl May 27 '19 at 00:15