Anti-Aircraft Marksman

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

Also at t=0,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 m/s,1000 \text{ m/s}, and it can be aimed in any direction. Your position is (x,y,z)=(0 m, 0 m, 0 m).(x,y,z) = (0 \text{ m, }0 \text{ m, }0\text{ m}). There is an ambient gravitational acceleration of 10 m/s210 \text{ m/s}^2 in the z-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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