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8. (R8M.9) Suppose that in a certain particle physics experiment, a particle of mass $m_0$ moving with speed $|\vec{v}_0| = \frac{3}{5}$ in the $+x$ direction in the laboratory frame is observed to decay into two particles, one with mass $m_1$ moving with speed $|\vec{v}_1| = \frac{4}{5}$ in the $+x$ direction and another with mass $m_2$ that is essentially at rest. (a) Show that if total particle mass and Newtonian momentum are conserved in the laboratory frame, then we must have $m_1 = \frac{3}{4}m_0$ and $m_2 = \frac{1}{4}m_0$. (b) Show that if four-momentum is to be conserved in the laboratory frame, we must have $m_1 = \frac{9}{16}m_0$ and $m_2 = \frac{5}{16}m_0$ (note that this decay converts some mass energy to kinetic energy if conservation of four-momentum is true). (c) Consider now an inertial reference frame (the Other Frame) that moves with the same speed in the $+x$ direction as the initial particle. Use the Einstein velocity transformation equations: $v_x' = \frac{v_x - \beta}{1 - \beta v_x}$, $v_y' = \frac{v_y}{\gamma (1 - \beta v_x)}$, $v_z' = \frac{v_z}{\gamma (1 - \beta v_x)}$ to show that the particles' $x$-velocities in this frame must be $v_{1x}' = 0$, $v_{1x}' = \frac{5}{13}$, and (d) Show that if we use the outgoing particle masses that ensure that Newtonian momentum and particle mass are conserved in the laboratory frame [see part (a)], then Newtonian momentum is not conserved in the Other Frame. (e) Show that if we use the masses that ensure that fourmomentum is conserved in the laboratory frame [see part (b)], we find that four-momentum is conserved in the Other Frame.

          8. (R8M.9) Suppose that in a certain particle physics experiment, a particle of mass $m_0$ moving with speed $|\vec{v}_0| = \frac{3}{5}$ in the $+x$ direction in the laboratory frame is observed to decay into two particles, one with mass $m_1$ moving with speed $|\vec{v}_1| = \frac{4}{5}$ in the $+x$ direction and another with mass $m_2$ that is essentially at rest.
(a) Show that if total particle mass and Newtonian momentum are conserved in the laboratory frame, then we must have $m_1 = \frac{3}{4}m_0$ and $m_2 = \frac{1}{4}m_0$.
(b) Show that if four-momentum is to be conserved in the laboratory frame, we must have $m_1 = \frac{9}{16}m_0$ and $m_2 = \frac{5}{16}m_0$ (note that this decay converts some mass energy to kinetic energy if conservation of four-momentum is true).
(c) Consider now an inertial reference frame (the Other Frame) that moves with the same speed in the $+x$ direction as the initial particle. Use the Einstein velocity transformation equations:
$v_x' = \frac{v_x - \beta}{1 - \beta v_x}$,
$v_y' = \frac{v_y}{\gamma (1 - \beta v_x)}$,
$v_z' = \frac{v_z}{\gamma (1 - \beta v_x)}$
to show that the particles' $x$-velocities in this frame must be $v_{1x}' = 0$, $v_{1x}' = \frac{5}{13}$, and
(d) Show that if we use the outgoing particle masses that ensure that Newtonian momentum and particle mass are conserved in the laboratory frame [see part (a)], then Newtonian momentum is not conserved in the Other Frame.
(e) Show that if we use the masses that ensure that fourmomentum is conserved in the laboratory frame [see part (b)], we find that four-momentum is conserved in the Other Frame.

8. (R8M.9) Suppose that in a certain particle physics experiment, a particle of mass m0 moving with speed |v⃗0| = (3)/(5) in the +x direction in the laboratory frame is observed to decay into two particles, one with mass m1 moving with speed |v⃗1| = (4)/(5) in the +x direction and another with mass m2 that is essentially at rest.
(a) Show that if total particle mass and Newtonian momentum are conserved in the laboratory frame, then we must have m1 = (3)/(4)m0 and m2 = (1)/(4)m0.
(b) Show that if four-momentum is to be conserved in the laboratory frame, we must have m1 = (9)/(16)m0 and m2 = (5)/(16)m0 (note that this decay converts some mass energy to kinetic energy if conservation of four-momentum is true).
(c) Consider now an inertial reference frame (the Other Frame) that moves with the same speed in the +x direction as the initial particle. Use the Einstein velocity transformation equations:
vx' = (vx - β)/(1 - β vx),
vy' = (vy)/(γ (1 - β vx)),
vz' = (vz)/(γ (1 - β vx))
to show that the particles' x-velocities in this frame must be v1x' = 0, v1x' = (5)/(13), and
(d) Show that if we use the outgoing particle masses that ensure that Newtonian momentum and particle mass are conserved in the laboratory frame [see part (a)], then Newtonian momentum is not conserved in the Other Frame.
(e) Show that if we use the masses that ensure that fourmomentum is conserved in the laboratory frame [see part (b)], we find that four-momentum is conserved in the Other Frame.

Added by Adriana C.

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