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Non-Newtonian Mechanics.

377

composition of velocities we find for these velocities the relations $u_{1}={\tfrac {u-v}{1-uv/c^{2}}}$ and $u_{2}={\tfrac {-u-v}{1+uv/c^{2}}}$ . Since these velocities are not of the same magnitude, the two bodies which have the same mass when at rest do not now have the same mass to this observer. Let us call these masses before collision $m_{1}$ and $m_{2}$ . During collision, the velocities of the bodies will all the time be changing; from the principle of the conservat1on of mass, however, the sum of the two masses will always equal $m_{1}+m_{2}$ ^[1]. When in the course of the collision the bodies have come to relative rest and are both moving past our observer with the velocity - $v$ , their momentum will be $-\left(m_{1}+m_{2}\right)v$ , and from the principle of the conservation of momentum this must be equal to the original momentum before collision, giving us the equation,-

-\left(m_{1}+m_{2}\right)v=m_{1}u_{1}+m_{2}u_{2}=m_{1}{\frac {u-v}{1-{\frac {uv}{c^{2}}}}}+m_{2}{\frac {-u-v}{1+{\frac {uv}{c^{2}}}}}.

(1)

Simplifying, we have,-

{\frac {m_{1}}{m_{2}}}={\frac {1-{\frac {uv}{c^{2}}}}{1+{\frac {uv}{c^{2}}}}},

(2)

which by direct algebraic transformations may be shown to be identical with

{\frac {m_{1}}{m_{2}}}={\frac {\sqrt {1-{\frac {\left({\frac {-u-v}{1+{\frac {uv}{c^{2}}}}}\right)^{2}}{c^{2}}}}}{\sqrt {1-{\frac {\left({\frac {u-v}{1-{\frac {uv}{c^{2}}}}}\right)^{2}}{c^{2}}}}}}={\frac {\sqrt {1-{\frac {u_{2}^{2}}{c^{2}}}}}{\sqrt {1-{\frac {u_{1}^{2}}{c^{2}}}}}}.

(3)

↑ In this connexion an interesting fact has been pointed out to the writer by Professor Lewis. As stated above, the sum of the two masses is throughout collision always equal to $m_{1}+m_{2}$ , and hence also at the time in the collision when the masses have come to relative rest their sum is $m_{1}+m_{2}$ . Since at this time both bodies are mowing with the velocity - $v$ we might suppose that $m_{1}+m_{2}$ equals $2m_{0}/{\sqrt {1-u^{2}/c^{2}}}$ . This is not the case, however, since the bodies now possess additional elastic energy beyond that which they possess when at rest and not in contact. A relation between mass and energy has already been developed(loc. cit.), and the mass of this elastic energy must also be taken into account in calculating $m_{1}+m_{2}$ . In fact the consideration of a collision of this type leads to a simple proof of the relation between mass and energy, a proof presented by Professor Lewis in a series of lectures on the Theory of Relativity given at Harvard University in the Spring of 1911.

[1] In this connexion an interesting fact has been pointed out to the writer by Professor Lewis. As stated above, the sum of the two masses is throughout collision always equal to $m_{1}+m_{2}$ , and hence also at the time in the collision when the masses have come to relative rest their sum is $m_{1}+m_{2}$ . Since at this time both bodies are mowing with the velocity - $v$ we might suppose that $m_{1}+m_{2}$ equals $2m_{0}/{\sqrt {1-u^{2}/c^{2}}}$ . This is not the case, however, since the bodies now possess additional elastic energy beyond that which they possess when at rest and not in contact. A relation between mass and energy has already been developed(loc. cit.), and the mass of this elastic energy must also be taken into account in calculating $m_{1}+m_{2}$ . In fact the consideration of a collision of this type leads to a simple proof of the relation between mass and energy, a proof presented by Professor Lewis in a series of lectures on the Theory of Relativity given at Harvard University in the Spring of 1911.

[1]