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183_notes:examples:maximally_inelastic_collision_of_two_identical_carts [2014/11/04 07:08] – pwirving | 183_notes:examples:maximally_inelastic_collision_of_two_identical_carts [2014/11/06 03:46] (current) – pwirving | ||
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=== Lacking === | === Lacking === | ||
+ | Find the final momentum, final speed, and final kinetic energy of the carts in terms of their initial values. | ||
+ | What is the change in internal energy of the two carts? | ||
=== Approximations & Assumptions === | === Approximations & Assumptions === | ||
- | Neglect | + | External forces are negligible during the collision, so neglect |
=== Representations === | === Representations === | ||
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Surroundings: | Surroundings: | ||
+ | {{183_notes: | ||
+ | |||
+ | $\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$ | ||
+ | |||
+ | $E_f = E_i + W + Q$ | ||
+ | |||
+ | $K = \frac{1}{2}mv^{2} = \frac{1}{2}m(\frac{p}{m})^{2} = \frac{1}{2}m(\frac{p^{2}}{m})$ | ||
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Since the y and z components of momentum don't change, we can work with only x components | Since the y and z components of momentum don't change, we can work with only x components | ||
- | From the momentum principle (x components): | + | From the momentum principle (x components) |
- | $$\vec{p}_{1xf} + \vec{p}_{2xf} = \vec{p}_{1xi}$$ | + | $${p}_{1xf} + {p}_{2xf} = {p}_{1xi}$$ |
+ | |||
+ | After the collision ${p}_{2xf}$ is equal to ${p}_{1xf}$ as they are stuck together so: | ||
$$2p_{1xf} = p_{1xi}$$ | $$2p_{1xf} = p_{1xi}$$ | ||
+ | |||
+ | Rearrange to isolate $p_{1xf}$ | ||
$$p_{1xf} = \dfrac{1}{2}p_{1xi}$$ | $$p_{1xf} = \dfrac{1}{2}p_{1xi}$$ | ||
- | The final speed of the stuck-together carts its half the initial speed: | + | Therefore the final speed of the stuck-together carts its half the initial speed: |
$$v_{f} = \dfrac{1}{2}{v_{i}}$$ | $$v_{f} = \dfrac{1}{2}{v_{i}}$$ | ||
- | Final translational kinetic energy: | + | Since we know the speed of the carts we can calculate their translational kinetic energy. |
+ | |||
+ | Final translational kinetic energy | ||
$$(K_{1f} + K_{2f}) = 2(\dfrac{1}{2}mv^2_{f})$$ | $$(K_{1f} + K_{2f}) = 2(\dfrac{1}{2}mv^2_{f})$$ | ||
+ | |||
+ | Substitute in the final speed of the stuck-together carts: | ||
$$(K_{1f} + K_{2f}) = 2(\dfrac{1}{2}m(\dfrac{1}{2}v_{i})^2) = \dfrac{1}{4}mv^2_{i}$$ | $$(K_{1f} + K_{2f}) = 2(\dfrac{1}{2}m(\dfrac{1}{2}v_{i})^2) = \dfrac{1}{4}mv^2_{i}$$ | ||
+ | Substituting back in $K_{1i}$ in order to put the final kinetic energy in terms of the initial kinetic energy we get: | ||
+ | |||
+ | $$(K_{1f} + K_{2f}) = \dfrac{K_{1i}}{2}$$ | ||
+ | |||
+ | We know the initial and final translational kinetic energies of the system, so we can use the energy principle to find the change in internal energy: | ||
+ | |||
+ | $$K_{1f} + K_{2f} + E_{int,f} = K_{1i} + E_{int,i}$$ | ||
+ | |||
+ | Rearrange to get the final internal energy minus the initial internal energy one one side. | ||
+ | |||
+ | $$E_{int,f} - E_{int,i} = K_{1i} - (K_{1f} + K_{2f})$$ | ||
+ | $E_{int,f} - E_{int,i}$ is the same as $\Delta E_{int}$ and this is what we are trying to find so substitute this in. Also substitute $\dfrac{K_{1i}}{2}$ for $(K_{1f} + K_{2f})$. We get: | ||
+ | $$\Delta E_{int} = K_{1i} - \dfrac{K_{1i}}{2}$$ | ||
+ | Resolve the right hand side and you get: | ||
+ | $$\Delta E_{int} = \dfrac{K_{1i}}{2}$$ | ||
+ | The final kinetic energy of the system is only half of the original kinetic energy, which means that the other half of the original kinetic energy has been dissipated into increased internal energy $\Delta E_{int}$ of the two carts. | ||
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- | $$E_f = E_i + W + Q$$ | ||
- | $$K_{1f} + K_{2f} + E_{int1f} + E_{int2f} = K_{1i} + K_{2i} + E_{int1i} + E_{int2i}$$ | ||
- | $$K_{1f} + K_{2f} = K_{1i}$$ | ||
- | Combine momentum and energy equations: | ||
- | $$\dfrac{p^{2}_{1xf}}{2m} + \dfrac{p^{2}_{2xf}}{2m} = \dfrac{(p_{1xf} + p_{2xf})^2}{2m}$$ | ||
- | $$p^{2}_{1xf} + p^{2}_{2xf} = p^{2}_{1xf} + 2{p_{1xf}p_{2xf}} + p^{2}_{2xf}$$ | ||
- | $$2{p_{1xf}p_{2xf}} = 0$$ | ||
- | There are two possible solutions to this equation. The term ${p_{1xf}p_{2xf}}$ can be zero if $p_{1xf} = 0$ or if $p_{2xf} = 0$. |