A heavy nucleus , at rest, undergoes fission where and are two lighter nuclei. Let where , and are the masses of , and , respectively. and are the kinetic energies of and , respectively. The speeds of and are and , respectively. If is the sp

Question

A heavy nucleus $N$ , at rest, undergoes fission $N \to P + Q,$ where $P$ and $Q$ are two lighter nuclei. Let $δ = M_{N} - M_{P} - M_{Q},$ where $M_{P}$ , $M_{Q}$ and $M_{N}$ are the masses of $P$ , $Q$ and $N$ , respectively. $E_{P}$ and $E_{Q}$ are the kinetic energies of $P$ and $Q$ , respectively. The speeds of $P$ and $Q$ are $v_{P}$ and $v_{Q}$ , respectively. If $c$ is the speed of light, which of the following statement(s) is(are) correct? [2021]

Smart Achievers · Accepted Answer

(1, 3, 4)

For nuclear fission reaction,

$N \to P + Q$

Energy released $= Δ M c^{2} = (M_{N} - M_{P} - M_{Q}) c^{2} = δ c^{2}$

This will be distributed as kinetic energy of $P$ and $Q$

$∴$ $E_{P} + E_{Q} = δ c^{2}$ $Q$

By conservation of momentum,

$V_{P} M_{P} = V_{Q} M_{Q}$

$∴$ $\frac{V_{P}}{V_{Q}} = \frac{M_{Q}}{M_{P}}$ $\dots (i i)$

Kinetic energy can be written as $KE = \frac{P^{2}}{2 m}$

Hence, kinetic energies are divided in the inverse ratio of masses.

$E_{P} = \frac{M_{Q}}{M_{P} + M_{Q}} δ c^{2}$ $\dots (i i i)$

By equation (i),

$\Rightarrow \frac{P^{2}}{2 M_{P}} + \frac{P^{2}}{2 M_{Q}} = δ c^{2}$

$\Rightarrow \frac{P^{2}}{2 μ} = δ c^{2}$ $\Rightarrow P = c \sqrt{2 μ δ}$ $[∵ μ = \frac{M_{P} M_{Q}}{M_{P} + M_{Q}}]$

Solution

1 $E_{P} + E_{Q} = c^{2} δ$

2 $E_{P} = (\frac{M_{P}}{M_{P} + M_{Q}}) c^{2} δ$

3 $\frac{v_{P}}{v_{Q}} = \frac{M_{Q}}{M_{P}}$

4 The magnitude of momentum for $P$ as well as $Q$ is $c \sqrt{2 μ δ},$ where $μ = \frac{M_{P} M_{Q}}{(M_{P} + M_{Q})}$

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Solution

1 EP+EQ=c2δ

2 EP=(MPMP+MQ)c2δ

3 vPvQ=MQMP

4 The magnitude of momentum for P as well as Q is c2μδ, where μ=MPMQ(MP+MQ)

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1 $E_{P} + E_{Q} = c^{2} δ$

2 $E_{P} = (\frac{M_{P}}{M_{P} + M_{Q}}) c^{2} δ$

3 $\frac{v_{P}}{v_{Q}} = \frac{M_{Q}}{M_{P}}$

4 The magnitude of momentum for $P$ as well as $Q$ is $c \sqrt{2 μ δ},$ where $μ = \frac{M_{P} M_{Q}}{(M_{P} + M_{Q})}$