Two particles of masses $m_1, m_2$ move with initial velocities $u_1$and $u_2$ On collision, one of the particles get excited to higher level, after absorbing energy $\varepsilon $. If final velocities of particles be $v_1$ and $v_2$ then we must have
Two particles of masses $m_1, m_2$ move with initial velocities $u_1$and $u_2$ On collision, one of the particles get excited to higher level, after absorbing energy $\varepsilon $. If final velocities of particles be $v_1$ and $v_2$ then we must have
- [AIPMT 2015]
- A
$\frac{1}{2}{m_1}{u_1}^2 + \frac{1}{2}{m_2}{u_2}^2 = \frac{1}{2}{m_1}{v_1}^2 + \frac{1}{2}{m_2}{v_2}^2 - \varepsilon $
- B
$\;\frac{1}{2}{m_1}{u_1}^2 + \frac{1}{2}{m_2}{u_2}^2 - \varepsilon = \frac{1}{2}{m_1}{v_1}^2 + \frac{1}{2}{m_2}{v_2}^2$
- C
$\;\frac{1}{2}{m_1}{u_1}^2 + \frac{1}{2}{m_2}{u_2}^2 + \varepsilon = \frac{1}{2}{m_1}{v_1}^2 + \frac{1}{2}{m_2}{v_2}^2$
- D
$m_1^2u_1+m_2^2u_2 - \varepsilon = m_1^2v_1+m_2^2v_2$
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$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
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$(a)$ Show $p = p _{t}^{\prime}+m_{t} V$
where $p$, is the momentum of the the particle (of mass $m$ ) and $p_{t}^{\prime \prime}=m_{t} v_{t}$,
Note $v_{t}$, is the velocity of the particle relative to the centre of mass. Also, prove using the definition of the centre of mass $\sum p _{t}^{\prime}=0$
$(b)$ Show $K=K^{\prime}+1 / 2 M V^{2}$
where $K$ is the total kinetic energy of the system of particles. $K^{\prime}$ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $M V^{2} / 2$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system).
$(c)$ Show $L = L ^{\prime}+ R \times M V$
where $L ^{\prime}=\sum r _{t}^{\prime} \times p _{t}^{\prime}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r _{t}^{\prime}= r _{t}- R$; rest of the notation is the standard notation used in the chapter. Note $L$ ' and $M R \times V$ can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
$(d)$ Show $\frac{d L ^{\prime}}{d t}=\sum r _{t}^{\prime} \times \frac{d p ^{\prime}}{d t}$
Further, show that
$\frac{d L ^{\prime}}{d t}=\tau_{e x t}^{\prime}$
where $\tau_{c t t}^{\prime}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
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