To solve the problem of the cannon recoiling after firing a cannonball, we will use the principle of conservation of momentum. Here’s a step-by-step solution:
### Step 1: Understand the Problem
We have a cannon of mass \( m_c = 150 \, \text{kg} \) and a cannonball of mass \( m_b = 1.5 \, \text{kg} \) that is shot with a velocity \( v_b = 60 \, \text{m/s} \). We need to find the recoil speed \( v_c \) of the cannon after firing.
### Step 2: Apply the Conservation of Momentum
According to the law of conservation of momentum, the total momentum before the event (firing the cannonball) must equal the total momentum after the event.
**Initial Momentum:**
Since both the cannon and cannonball are at rest before firing, the initial momentum \( P_{\text{initial}} \) is:
\[
P_{\text{initial}} = 0
\]
**Final Momentum:**
After the cannonball is fired, the momentum of the cannonball and the cannon must equal the initial momentum:
\[
P_{\text{final}} = m_b \cdot v_b + m_c \cdot v_c
\]
Where:
- \( m_b \) is the mass of the cannonball,
- \( v_b \) is the velocity of the cannonball,
- \( m_c \) is the mass of the cannon,
- \( v_c \) is the recoil velocity of the cannon (which we need to find).
### Step 3: Set Up the Equation
Since momentum is conserved:
\[
P_{\text{initial}} = P_{\text{final}}
\]
This gives us:
\[
0 = m_b \cdot v_b + m_c \cdot v_c
\]
### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[
0 = (1.5 \, \text{kg}) \cdot (60 \, \text{m/s}) + (150 \, \text{kg}) \cdot v_c
\]
\[
0 = 90 \, \text{kg m/s} + 150 \, \text{kg} \cdot v_c
\]
### Step 5: Solve for \( v_c \)
Rearranging the equation to solve for \( v_c \):
\[
150 \, v_c = -90
\]
\[
v_c = \frac{-90}{150}
\]
\[
v_c = -0.6 \, \text{m/s}
\]
### Step 6: Interpret the Result
The negative sign indicates that the cannon recoils in the opposite direction to the cannonball. Thus, the speed of the cannon's recoil is \( 0.6 \, \text{m/s} \).
### Final Answer
The recoil speed of the cannon is \( 0.6 \, \text{m/s} \) in the direction opposite to that of the cannonball.
---
