Suppose that a mass M1 is moving with speed V1 and collides with mass M2 which is initially at rest. After the elastic collision they make, both momentum and kinetic energy are conserved.
[tex] m_{1}v_{1f} + m_{2}v_{2f} = m_{1}v_{1i} [/tex]
[tex] \frac{1}{2}m_{1}||v_{1i}||^{2}= \frac{1}{2}m_{1}||v_{1f}||^{2} + \frac{1}{2}m_{2}||v_{2f}||^{2} [/tex]
Derive the following equations:
[tex] v_{1f} = \frac{m_{1}-m_{2}}{m_{1}+m_{2}}v_{1i} [/tex]
[tex] v_{2f} = \frac{2m_{1}}{m_{1}+m_{2}}v_{1i} [/tex]
Resnick & Halliday give a fairly staightfoward proof. But in the proof it fails to recognize the fact that the v values in the momentum conservation are vectors, whereas those in the energy conservation are scalars. So the proof is not rigorous. I was curious how one would prove this rigorously, (preferably without casework), given this remark.
Thanks!
BiP
[tex] m_{1}v_{1f} + m_{2}v_{2f} = m_{1}v_{1i} [/tex]
[tex] \frac{1}{2}m_{1}||v_{1i}||^{2}= \frac{1}{2}m_{1}||v_{1f}||^{2} + \frac{1}{2}m_{2}||v_{2f}||^{2} [/tex]
Derive the following equations:
[tex] v_{1f} = \frac{m_{1}-m_{2}}{m_{1}+m_{2}}v_{1i} [/tex]
[tex] v_{2f} = \frac{2m_{1}}{m_{1}+m_{2}}v_{1i} [/tex]
Resnick & Halliday give a fairly staightfoward proof. But in the proof it fails to recognize the fact that the v values in the momentum conservation are vectors, whereas those in the energy conservation are scalars. So the proof is not rigorous. I was curious how one would prove this rigorously, (preferably without casework), given this remark.
Thanks!
BiP