Challenging Applied maths question projectiles

[Lukesean]

An aircraft flies at a constant height H and constant velocity V. When the aircraft has flown directly over a gun on the ground a shot is fired from the gun which points at the aircraft at an angle of elevation Ѳ. If the initial velocity of the bullet is KVsecѲ [k >1], and Ѳ = tan^-11/v[ _√gh/(k-1)], show that the bullet hits the aircraft directly.

 

[Quaid]

An aircraft flies at a constant height H and constant velocity V. When the aircraft has flown directly over a gun on the ground a shot is fired from the gun which points at the aircraft at an angle of elevation Ѳ. If the initial velocity of the bullet is KVsecѲ [k >1], and Ѳ = tan^-11/v[ _√gh/(k-1)], show that the bullet hits the aircraft directly.

Hi Lukesean:

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Please explain what you've done and thought about in this exercise, so far.

Also, the expression highlighted in red above is ambiguous. Please retype the expression for theta, using grouping symbols to (1) clearly show the argument for arctan and (2) clearly show what part is the radicand, and also (3) explain whether symbols v and V are supposed to represent the same quantity.

Thank you!

 

[wjm11]

An aircraft flies at a constant height H and constant velocity V. When the aircraft has flown directly over a gun on the ground a shot is fired from the gun which points at the aircraft at an angle of elevation Ѳ. If the initial velocity of the bullet is KVsecѲ [k >1], and Ѳ = tan^-11/v[ _√gh/(k-1)], show that the bullet hits the aircraft directly.

In general, since we are not considering drag effects on the projectile, only two things are necessary to guarantee a hit: 1) The horizontal component of the projectile velocity is equal to the aircraft V, and 2) The initial vertical component of velocity is sufficient to reach the height of the aircraft.

Hope that helps.