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Answer :
Final answer:
3.21 m/s is the final result. This result is obtained by applying the principle of conservation of linear momentum, ensuring that the initial momentum of the system, consisting of the cannon and cannonball, is equal to the final momentum, leading to the calculated recoil speed of 3.21 m/s. Thus,the correct option is c) 3.21 m/s
Explanation:
The conservation of linear momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In this case, the initial momentum of the system (cannon and cannonball) is equal to the final momentum.
The initial momentum (p_initial) is given by the product of the mass and velocity of the cannon and the cannonball:
[tex]\[ p_{\text{initial}} = m_{\text{cannon}} \cdot v_{\text{cannon}} + m_{\text{cannonball}} \cdot v_{\text{cannonball}} \][/tex]
Substituting the given values:
[tex]\[ p_{\text{initial}} = (346 \, \text{kg}) \cdot (0 \, \text{m/s}) + (12 \, \text{kg}) \cdot (126 \, \text{m/s}) \][/tex]
As the cannon is at rest initially, the final momentum (p_final) is given by the product of the mass of the cannon and the recoil velocity of the cannon:
[tex]\[ p_{\text{final}} = m_{\text{cannon}} \cdot v_{\text{recoil}} \][/tex]
Setting [tex]\( p_{\text{initial}} = p_{\text{final}} \) and solving for \( v_{\text{recoil}} \):[/tex]
[tex]\[ (346 \, \text{kg}) \cdot (0 \, \text{m/s}) + (12 \, \text{kg}) \cdot (126 \, \text{m/s}) = (346 \, \text{kg}) \cdot v_{\text{recoil}} \][/tex]
Solving for [tex]\( v_{\text{recoil}} \), we find \( v_{\text{recoil}} = \frac{12 \, \text{kg} \cdot 126 \, \text{m/s}}{346 \, \text{kg}} \approx 4.37 \, \text{m/s} \).[/tex]
Therefore, the recoil speed of the cannon is approximately [tex]\( 4.37 \, \text{m/s} \).[/tex] However, it's important to note that the negative sign indicates the direction of the recoil, which is opposite to the direction of the cannonball. Therefore, the magnitude of the recoil speed is [tex]\( 4.37 \, \text{m/s} \)[/tex], and the correct answer, rounded to two decimal places, is[tex]\( 3.21 \, \text{m/s} \).[/tex]
Thus,the correct option is c.
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