Question

What was the speed of the cannon ball if the spring compressed 58 cm when the cannon was fired?

Answer 1

I attached the full question.
We need to write down the law of conservation of momentum and energy.
The law of conservation of momentum:
$m_cv_c=m_bv_b$
The law of conservation of energy:
$\frac{1}{2}x^2=\frac{m_c v_c^2}{2}$
This tells us that potential energy of the compressed spring is used to stop the canon. In other words, the kinetic energy of the canon, after firing is used to do work again the force of the spring.
We can use this two equations to find the velocity of the cannonball:
$\frac{1}{2}x^2=\frac{m_c v_c^2}{2}\\ v_c=\sqrt{\frac{1}{m_c}}}\cdot x$
We can now plug this in the first equation:
$m_cv_c=m_bv_b\\ v_c=\sqrt{\frac{1}{m_c}}}\cdot x\\ m_c\cdot\sqrt{\frac{1}{m_c}}}\cdot x=m_bv_b\\ \sqrt{m_c}}}\cdot x=m_bv_b\\ v_b=\frac{\sqrt{m_c}x}{m_b}$
Please note that I found this question online with multiple diferent data( mass of the canon, ball etc)
You can use this formula, just plug in your numbers and you will get the correct answer.