To solve the problem step by step, we will use the principle of conservation of momentum. The initial momentum of the system is zero since the shell is at rest. After the explosion, the total momentum in both the x and y directions must also equal zero.
### Step 1: Identify the masses and their velocities
- Mass of fragment 1 (m1) = 1 kg, velocity (v1) = 12 m/s (along x-axis)
- Mass of fragment 2 (m2) = 2 kg, velocity (v2) = 8 m/s (along y-axis)
- Mass of fragment 3 (m3) = m kg, velocity (v3) = 40 m/s (direction to be determined)
### Step 2: Calculate the momentum of the first two fragments
- Momentum of fragment 1 (P1) = m1 * v1 = 1 kg * 12 m/s = 12 kg·m/s (along x-axis)
- Momentum of fragment 2 (P2) = m2 * v2 = 2 kg * 8 m/s = 16 kg·m/s (along y-axis)
### Step 3: Set up the conservation of momentum equations
Since the initial momentum is zero, the total momentum in both the x and y directions after the explosion must also be zero.
#### For the x-direction:
\[ P_{initial,x} = P_{final,x} \]
\[ 0 = P1 + P3_x \]
Where \( P3_x \) is the x-component of the momentum of the third fragment (m kg).
Since fragment 3 has no x-component of velocity, we can assume:
\[ P3_x = -P1 = -12 \, \text{kg·m/s} \]
#### For the y-direction:
\[ P_{initial,y} = P_{final,y} \]
\[ 0 = P2 + P3_y \]
Where \( P3_y \) is the y-component of the momentum of the third fragment (m kg).
Thus:
\[ P3_y = -P2 = -16 \, \text{kg·m/s} \]
### Step 4: Relate the momentum of fragment 3 to its speed
The momentum of fragment 3 can be expressed as:
\[ P3 = m \cdot v3 \]
Where \( v3 = 40 \, \text{m/s} \).
The total momentum of fragment 3 can be expressed in terms of its components:
\[ P3 = \sqrt{P3_x^2 + P3_y^2} \]
Substituting the values we found:
\[ P3 = \sqrt{(-12)^2 + (-16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \, \text{kg·m/s} \]
### Step 5: Set the momentum of fragment 3 equal to its mass times velocity
Now we can set up the equation:
\[ m \cdot 40 = 20 \]
Solving for m:
\[ m = \frac{20}{40} = 0.5 \, \text{kg} \]
### Step 6: Calculate the total mass of the shell
The total mass of the shell is the sum of the masses of the three fragments:
\[ \text{Total mass} = m1 + m2 + m3 = 1 \, \text{kg} + 2 \, \text{kg} + 0.5 \, \text{kg} = 3.5 \, \text{kg} \]
### Final Answer
The total mass of the shell is **3.5 kg**.
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