One type of decay seen in some subatomic particles occurs when a \'parent\' part

Question

One type of decay seen in some subatomic particles occurs when a 'parent' particle (mass M) spontaneously breaks up into two smaller 'daughter' particles (masses m_1 m_2). If the parent is initially at rest, then momentum is conserved during this decay as the daughters move in opposite directions with equal magnitudes of momentum: p_1 - p_2. Energy is also conserved: although the parent is not moving, it still has rest energy due to its mass (E_R - Mc^2), and all of this is used to create the two daughters: E_R - E_1 + E_2.. But the daughters are both moving, so some of the parent's rest (mass) energy is converted to kinetic energy. The result is that the total mass of the daughters is smaller than the mass of the parent: m_1 + m_2 0.90c for both daughters o second decay: choose any two speeds you like, so long as they average out to the speed used in the first decay e.g., if you were choosing car speeds in mph, for the first case you could choose 100 mph for both, and in the second case you could choose 50 mph and 150 mph, which average out to 100 mph. Or you could choose 120 mph and 80 mph. And so forth.

Explanation / Answer

when they have same speed:
let v1=v2=v=0.95*c

then gamma1=gamma2=1/sqrt(1-(v/c)^2)

=1/sqrt(1-0.95^2)

=3.2026

using conservation of momentum:

gamma1*m1*v1=gamma2*m2*v2

==>m1=m2

using energy conservation principle:

gamma1*m1*c^2+gamma2*m2*c^2=M*c^2

==>M=3.2026*(m1+m2)

==>m1+m2=M/3.2026

then lost mass=M-(m1+m2)

=0.68775*M

when they have separate speed:

let v1=0.96*c==>gamma1=1/sqrt(1-0.96^2)=3.5714

v2=0.94*c==>gamma2=1/sqrt(1-0.94^2)=2.9311

conserving momentum:

gamma1*m1*v1=gamma2*m2*v2

==>3.5714*m1*0.96*c=2.9311*m2*0.94*c

==>m2=1.24444*m1...(1)

conserving energy:

M*c^2=3.5714*m1*c^2+2.9311*m2*c^2

using equation 1,

M*c^2=3.5714*m1*c^2+2.9311*1.24444*m1*c^2=7.219*m1*c^2

==>M=7.219*m1...(2)

mass lost=M-(m1+m2)

=7.219*m1-m1-1.24444*m1

=4.9746*m1

=4.9746*M/7.219=0.6891*M

so mass lost is higher when they have separate speeds.

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