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Chapter 8: Problem 53

The USS Montana is a massive battleship with a weight of \(136,634,000 \mathrm{lb}\). It has twelve 16 -inch guns, which are capable of firing 2700 -lb projectiles at a speed of \(2300 \mathrm{ft} / \mathrm{s}\). If the battleship fires three of these guns (in the same direction), what is the recoil velocity of the ship?

Short Answer

Expert verified
Answer: The recoil velocity of the USS Montana battleship is approximately -0.161 ft/s. The negative sign indicates that the ship moves in the opposite direction of the fired projectiles.

Step by step solution

01

Write down the conservation of momentum equation

The conservation of momentum equation is given as: \(\text{momentum}_\text{before} = \text{momentum}_\text{after}\) In our case, since the ship is initially at rest, we have: \(\text{momentum of the ship}_\text{before} + \text{momentum of the projectiles}_\text{before} = \text{momentum of the ship}_\text{after} + \text{momentum of the projectiles}_\text{after}\)
02

Calculate the momentum of the projectiles before and after firing

Projectile mass: \(m_p=2700\mathrm{lb}\) Projectile velocity: \(v_p=2300\mathrm{ft/s}\) Number of projectiles fired: \(n=3\) Momentum of the projectiles before firing: \(\text{momentum}_\text{before(projectiles)}=0\) Momentum of the projectiles after firing: \(\text{momentum}_\text{after(projectiles)} = n \times m_p \times v_p\)
03

Calculate the momentum of the ship before and after firing

Ship mass: \(m_s=136,634,000\mathrm{lb}\) Recoil velocity of the ship: \(v_s\) (unknown) Momentum of the ship before firing: \(\text{momentum}_\text{before(ship)}=0\) Momentum of the ship after firing: \(\text{momentum}_\text{after(ship)}=m_s \times v_s\) Now we substitute the values from Steps 2 and 3 in the conservation of momentum equation.
04

Solve for the recoil velocity of the ship

Conservation of momentum equation: \(0 + 0 = m_s \times v_s + n \times m_p \times v_p\) Solve for \(v_s\): \(v_s = - \frac{n \times m_p \times v_p}{m_s}\) Now substitute the values of \(n\), \(m_p\), \(v_p\), and \(m_s\): \(v_s = - \frac{3 \times 2700 \mathrm{lb} \times 2300 \mathrm{ft/s}}{136,634,000 \mathrm{lb}}\) Calculate the value of \(v_s\): \(v_s \approx -0.161 \mathrm{ft/s}\)
05

Write down the final answer

The recoil velocity of the USS Montana battleship when it fires three of its guns in the same direction is approximately \(-0.161\mathrm{ft/s}\). The negative sign indicates that the ship moves in the opposite direction of the fired projectiles.

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