Analysis of the System

First, we calculate the properties of the Center of Mass (COM). The velocity of the COM remains constant because there are no external forces acting on the system.

Total Mass: $M = 30 \text{ kg}$
Fragment Mass: $m = 10 \text{ kg}$

Velocity of Center of Mass ($\vec{V}_{cm}$):

$$ \vec{V}_{cm} = \frac{m\vec{v}_1 + m\vec{v}_2 + m\vec{v}_3}{3m} = \frac{\vec{v}_1 + \vec{v}_2 + \vec{v}_3}{3} $$ $$ \vec{V}_{cm} = \frac{(3\hat{i} + 2\hat{j}) + (-2\hat{i}) + (5\hat{i} – 5\hat{j})}{3} $$ $$ \vec{V}_{cm} = \frac{6\hat{i} – 3\hat{j}}{3} = 2\hat{i} – \hat{j} \, \text{m/s} $$

Kinetic Energy of Center of Mass ($KE_{cm}$):

$$ |\vec{V}_{cm}|^2 = 2^2 + (-1)^2 = 5 $$ $$ KE_{cm} = \frac{1}{2} M V_{cm}^2 = \frac{1}{2}(30)(5) = 75 \text{ J} $$

Question 28: Energy Released in the Explosion

The energy released in the explosion is converted into the “internal” kinetic energy of the system, which is the kinetic energy of the fragments relative to the center of mass. Alternatively, it can be calculated as the Total Final Kinetic Energy minus the Initial Kinetic Energy (of the unexploded shell).

Since the problem asks for the energy released converted into KE, we calculate the total kinetic energy immediately after the explosion and subtract the translational KE of the COM (which represents the initial KE if the shell was just moving as a single body with momentum conservation).

Total Kinetic Energy ($KE_{\text{total}}$):

Energy Released ($E_{\text{rel}}$):

$$ E_{\text{rel}} = KE_{\text{total}} – KE_{cm} $$ $$ E_{\text{rel}} = 335 \text{ J} – 75 \text{ J} = 260 \text{ J} $$

Question 29: Minimum Total Kinetic Energy

We need to find the minimum total kinetic energy of the fragments after a sufficiently long time when the forces of mutual interaction cease to act.

Concepts:

• Energy Conservation: $E_{\text{total}} = KE + PE = \text{constant}$.
• Decomposition of KE: $KE_{\text{total}} = KE_{cm} + KE_{\text{relative}}$.
• $KE_{cm}$ is constant at 75 J.
• $KE_{\text{relative}} \ge 0$.

Analysis:

The problem states the forces “cease to act,” implying the particles separate to a very large distance ($r \to \infty$). Since the forces are attractive, the Potential Energy ($PE$) increases (approaches 0 from a negative value) as they separate. By conservation of energy, as $PE$ increases, $KE$ must decrease.

The question asks for the “minimum” total kinetic energy. The mathematical lower bound for total kinetic energy is $KE_{cm}$ (which occurs if all fragments move with the same velocity $\vec{V}_{cm}$, i.e., $KE_{\text{relative}} = 0$).

For the fragments to escape to infinity (“forces cease”), they must have enough energy. In the limiting case where they just barely escape the attractive potential well, their relative velocities at infinity will approach zero. In this specific limiting case:

$$ KE_{\text{relative}} \to 0 $$ $$ KE_{\text{total}} \to KE_{cm} $$

Thus, the minimum possible value for the total kinetic energy is the kinetic energy of the center of mass itself.