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Warships abreast, fewer than 300 yards apart, were in the firing position Lord Nelson's crews called "point-blank." But Nelson preferred his ship to be closer, "as close as possible" for from close range, a shot could go "through and through." That is, the shot could go through the close side of the enemy, inside across its deck then through the other side into the sea beyond.
Suppose Ship A fires a cannon, aimed horizontal, at Ship B. The shot mass is 24 pounds. Assume its muzzle speed constant at 500 ft/s over its flight. What does Newton's Second Law tell us about "Point Blank?"
The origin is the muzzle of the firing cannon. The extrinsic properties of the shot an instant before it impacts the ship (t = t impact-minus) are its position, velocity, and kinetic energy. The momentum equation with its initial and final conditions is given.
Solve the momentum equation to determine the velocity, speed and momentum of the shot an instant before its impact with Ship B.
♦ Position: We set our coordinate origin to be at the cannon muzzle in Ship A (See figure). Both ships move at near equal speeds on parallel courses. The differential equation and initial conditions for this event are:
The above equation integrates twice to yield:
With conditions applied, the equation becomes:
Separate the equation into I and K components then solve.
Thus after 0.48 seconds, the shot would be about to impact Ship B, 3.7 feet below the bore line of its cannons (assuming near identical ships).
♦ Velocity: A single integration of the momentum equation gives:
Inserting the time "instant prior to impact," 0.48 s, the velocity just prior to impact is:
♦ Kinetic Energy: In coming to rest after impact with the ship, nearly the entire kinetic energy of the shot will be transfered to the target ship. That amount of energy is:
Since I wrote this problem a few persons have told me that Nelson indeed coined the term "point blank" but that the distance abreast he sought to have between Victory and the enemy ship was less than three meters. Another term coined regarding those sea battles was "through and through." This described one effect of a shot (cannon ball) as being able to pass "though" the near hull of the enemy ship, zoom across its gun deck, then to pass "through" the other wall of the hull. Thus the shot passed "through and through." We use this expression today.