To solve the problem of calculating the recoil velocity of the gun when it fires a shell, we will use the principle of conservation of momentum. Here is the step-by-step solution:
### Step 1: Understand the Problem
We have a field gun of mass \( M = 1.5 \) tonnes and a shell of mass \( m = 15 \) kg. The shell is fired with a velocity \( u = 150 \) m/s. We need to find the recoil velocity \( V \) of the gun.
### Step 2: Convert Mass to Kilograms
Since the mass of the gun is given in tonnes, we need to convert it to kilograms:
\[
M = 1.5 \, \text{tonnes} = 1.5 \times 1000 \, \text{kg} = 1500 \, \text{kg}
\]
### Step 3: Write the Conservation of Momentum Equation
According to the law of conservation of momentum, the total momentum before firing is equal to the total momentum after firing. Initially, both the gun and the shell are at rest, so their initial momentum is zero:
\[
\text{Initial Momentum} = 0
\]
After firing, the momentum of the shell and the momentum of the gun must equal zero:
\[
m \cdot u + M \cdot V = 0
\]
Where:
- \( m \) is the mass of the shell,
- \( u \) is the velocity of the shell,
- \( M \) is the mass of the gun,
- \( V \) is the recoil velocity of the gun.
### Step 4: Rearrange the Equation
From the equation \( m \cdot u + M \cdot V = 0 \), we can express \( V \):
\[
M \cdot V = -m \cdot u
\]
\[
V = -\frac{m \cdot u}{M}
\]
### Step 5: Substitute the Values
Now we can substitute the known values into the equation:
\[
V = -\frac{15 \, \text{kg} \cdot 150 \, \text{m/s}}{1500 \, \text{kg}}
\]
### Step 6: Calculate the Recoil Velocity
Calculating the right-hand side:
\[
V = -\frac{2250 \, \text{kg m/s}}{1500 \, \text{kg}} = -1.5 \, \text{m/s}
\]
### Step 7: Interpret the Result
The negative sign indicates that the direction of the recoil velocity is opposite to the direction of the shell's velocity. Therefore, the recoil velocity of the gun is:
\[
V = 1.5 \, \text{m/s} \, \text{(in the opposite direction)}
\]
### Final Answer
The recoil velocity of the gun is \( 1.5 \, \text{m/s} \) in the opposite direction of the shell.
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