Question 23: Centre of Mass and Rotational Motion

Question

A neutron makes a head on elastic collision with a lead nucleus. The ratio of nuclear mass to neutron mass is 206 . The fractional change in kinetic energy of a neutron is

Accepted Answer

When a neutron undergoes a head-on elastic collision with a lead nucleus, we can analyze the process using the principles of conservation of momentum and kinetic energy, which apply since it's an elastic collision. The question involves calculating the fractional change in the kinetic energy of the neutron after the collision.
Let's denote:
$m_n$ as the mass of the neutron,
$m_p$ as the mass of the lead nucleus, where given it is 206 times the mass of the neutron, so $m_p = 206m_n$,
$v_i$ as the initial velocity of the neutron before the collision,
$v_f$ as the final velocity of the neutron after the collision,
$u_i = 0$ as the initial velocity of the lead nucleus (since it's considered stationary before the collision),
$u_f$ as the final velocity of the lead nucleus after the collision.
Since momentum is conserved in an elastic collision, the total initial momentum equals the total final momentum, which can be expressed as:
$m_n v_i + m_p u_i = m_n v_f + m_p u_f$
Given $u_i = 0$, we can simplify this to:
$m_n v_i = m_n v_f + m_p u_f$
Because the collision is elastic, kinetic energy is also conserved. The change in kinetic energy we are interested in involves calculating the kinetic energy before and after the collision, specifically for the neutron, and then finding the fractional change.
To find the velocities after the collision, we use the formula derived for velocities in a one-dimensional elastic collision involving two masses:
The velocity of the neutron ($v_f$) after the collision is given by:
$v_f = \frac{(m_n - m_p) v_i}{m_n + m_p}$
Substituting $m_p = 206m_n$, we get:
$v_f = \frac{(m_n - 206m_n) v_i}{m_n + 206m_n}$
$v_f = \frac{-205m_n}{207m_n} v_i$
$v_f = -\frac{205}{207} v_i$
The fractional change in kinetic energy of the neutron is given by the change in kinetic energy relative to the initial kinetic energy. The kinetic energy ($KE$) formula is $KE = \frac{1}{2}mv^2$, so the fractional change in kinetic energy for the neutron is:
$\Delta KE = \frac{\frac{1}{2}m_n v_f^2 - \frac{1}{2}m_n v_i^2}{\frac{1}{2}m_n v_i^2}$
Substituting $v_f = -\frac{205}{207} v_i$, we have:
$\Delta KE = \frac{\frac{1}{2}m_n (-\frac{205}{207} v_i)^2 - \frac{1}{2}m_n v_i^2}{\frac{1}{2}m_n v_i^2}$
=$\frac{\frac{1}{2}m_n (\frac{205}{207})^2 v_i^2 - \frac{1}{2}m_n v_i^2}{\frac{1}{2}m_n v_i^2}$
=$\frac{(\frac{205}{207})^2 - 1}{1}$
Calculating this gives us a fractional change in the kinetic energy:
=$\left(\frac{205^2 - 207^2}{207^2}\right)$
=$\left(\frac{42025 - 42849}{42849}\right)$
=$\frac{-824}{42849}$
This fraction simplifies approximately to a fractional decrease, and when calculated, it yields a value around -0.01923, which is about a 2% decrease in kinetic energy. Hence, the correct answer is:
Option B: 2% decrease.

Answer

3% increase

Answer

2% decrease

Answer

2% increase

Answer

3% decrease

Centre of Mass and Rotational Motion

A neutron makes a head on elastic collision with a lead nucleus. The ratio of nuclear mass to neutron mass is 206 . The fractional change in kinetic energy of a neutron is

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