Anti-Aircraft Marksman

You are a soldier, and your base is under attack by enemy aircraft. At time \(t = 0\), one such aircraft is positioned at \((x,y,z) = (1000\text{ m, }2000 \text{ m, }3000 \text{ m}).\) The aircraft has a constant velocity of \((v_x,v_y,v_z) = (-400 \text{ m/s, }+400 \text{ m/s, }+400 \text{ m/s}).\)

Also at \(t = 0,\) you take control of an anti-aircraft artillery cannon and fire a shot to bring down the enemy plane. The cannon has a muzzle velocity of \(1000 \text{ m/s},\) and it can be aimed in any direction. Your position is \((x,y,z) = (0 \text{ m, }0 \text{ m, }0\text{ m}).\)

There is an ambient gravitational acceleration of \(10 \text{ m/s}^2\) in the \(-z\) direction (downward).

At what time (in seconds) does the artillery shell strike the enemy plane?

Details and Assumptions:

  • Neglect air resistance.
  • Give your answer to 2 decimal places.
  • From a targeting perspective, consider the enemy plane to be a point-particle.
  • You are a superlative marksman.
  • If you find more than one solution, give the smaller value (provided that it's not negative).

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