To solve this problem, we can apply the principle of conservation of momentum. This principle states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces are acting on it. In this case, the cannon and the ball form a closed system during the firing event.
Understanding the System
Initially, both the cannon and the ball are at rest, which means their initial velocities are zero. When the cannon fires the ball, the cannon recoils backward, and the ball moves forward. We need to find the velocity of the recoiling cannon (V) and the initial velocity of the ball with respect to Earth (vE).
Applying Conservation of Momentum
The total momentum before firing is zero since both the cannon and the ball are stationary. After the cannon fires the ball, the momentum of the system can be expressed as:
- Momentum of the cannon: M * V
- Momentum of the ball: m * vE
Where:
- M = mass of the cannon = 1300 kg
- m = mass of the ball = 72 kg
- v = speed of the ball relative to the cannon = 55 m/s
Finding the Velocity of the Ball with Respect to Earth
Since the ball is fired at a speed of 55 m/s relative to the cannon, we can express the velocity of the ball with respect to Earth (vE) as:
vE = V + v
Setting Up the Equation
Using the conservation of momentum, we set up the equation:
0 = M * V + m * vE
Substituting vE into the momentum equation gives:
0 = M * V + m * (V + v)
Solving for the Recoiling Cannon's Velocity
Now, let's substitute the known values into the equation:
0 = 1300 * V + 72 * (V + 55)
Expanding this, we get:
0 = 1300V + 72V + 3960
Combining like terms results in:
0 = (1300 + 72)V + 3960
0 = 1372V + 3960
Now, we can isolate V:
1372V = -3960
V = -3960 / 1372
Calculating this gives:
V ≈ -2.89 m/s
Finalizing the Ball's Velocity
Now that we have the velocity of the cannon, we can find the velocity of the ball with respect to Earth:
vE = V + v = -2.89 + 55
vE ≈ 52.11 m/s
Summary of Results
To summarize:
- The velocity of the recoiling cannon (V) is approximately -2.89 m/s (indicating it moves in the opposite direction of the ball).
- The initial velocity of the ball with respect to Earth (vE) is approximately 52.11 m/s.
This example illustrates how momentum conservation can be applied to analyze the motion of objects in a system, even when they interact dynamically, like in the case of a cannon firing a projectile.
