To solve the problem step by step, we will use the principles of conservation of momentum and kinetic energy.
### Step 1: Understand the Problem
We have a bomb of mass \(3.0 \, \text{kg}\) that explodes into two pieces: one of mass \(2.0 \, \text{kg}\) and the other of mass \(1.0 \, \text{kg}\). The smaller mass (1.0 kg) moves at a speed of \(80 \, \text{m/s}\). We need to find the total energy imparted to the two fragments.
### Step 2: Apply Conservation of Momentum
Since no external forces are acting on the system, the momentum before the explosion must equal the momentum after the explosion. Initially, the bomb is at rest, so the initial momentum is \(0\).
Let:
- \(m_1 = 1.0 \, \text{kg}\) (smaller mass)
- \(m_2 = 2.0 \, \text{kg}\) (larger mass)
- \(v_1 = 80 \, \text{m/s}\) (velocity of smaller mass)
- \(v_2\) = velocity of larger mass (to be determined)
Using the conservation of momentum:
\[
0 = m_1 v_1 + m_2 v_2
\]
Substituting the known values:
\[
0 = (1.0 \, \text{kg})(80 \, \text{m/s}) + (2.0 \, \text{kg}) v_2
\]
\[
0 = 80 + 2 v_2
\]
Solving for \(v_2\):
\[
2 v_2 = -80 \implies v_2 = -40 \, \text{m/s}
\]
The negative sign indicates that the larger mass moves in the opposite direction to the smaller mass.
### Step 3: Calculate the Kinetic Energy of Each Fragment
The kinetic energy (KE) of an object is given by the formula:
\[
KE = \frac{1}{2} m v^2
\]
**For the smaller mass:**
\[
KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} (1.0 \, \text{kg}) (80 \, \text{m/s})^2
\]
\[
KE_1 = \frac{1}{2} (1.0) (6400) = 3200 \, \text{J}
\]
**For the larger mass:**
\[
KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (2.0 \, \text{kg}) (40 \, \text{m/s})^2
\]
\[
KE_2 = \frac{1}{2} (2.0) (1600) = 1600 \, \text{J}
\]
### Step 4: Calculate the Total Energy
The total energy imparted to the two fragments is the sum of their kinetic energies:
\[
\text{Total Energy} = KE_1 + KE_2 = 3200 \, \text{J} + 1600 \, \text{J} = 4800 \, \text{J}
\]
### Step 5: Convert to Kilojoules
To convert joules to kilojoules, divide by \(1000\):
\[
\text{Total Energy in kJ} = \frac{4800 \, \text{J}}{1000} = 4.8 \, \text{kJ}
\]
### Final Answer
The total energy imparted to the two fragments is \(4.8 \, \text{kJ}\).
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