A particle is aimed at a mark, which is in the horizontal plane through the point of projection, falls a meters short of it when the elevation is a and goes beyond it b meters too far when the elevation is 3. Show that if the velocity of projection in each case be the same, then the proper elevation, 0, is a sin 28 +b sin 2a a +b 1 0 = =s Question 2: (a) A projectile started from O at an elevation a. After t seconds its position appeared to have an elevation 3 as seen from O. Prove that the initial velocity was gt cos B 2sin (a – B) (b) A stone is projected with velocity v and elevation 0 from a point O on level ground so as to hit a mark P on a wall whose distance from O is a the height of P above the ground being b. Prove that 20 (a sin 0 cos 0 – b cos² 0) = ga² Also prove that the requisite velocity of projection is least when 0 = +;, where a is the elevation of P as seen from O.

The space shuttle uses radar to determine the magnitudes and direction cosines of position vectors to satellites A and B as for TOA: TOA ОА = 2 km, cos (0x) = 0.768, cos (y) = -0.384, cos (0) = 0.512, for OB: O | OB|= = 2.5 km, cos (0x) = 0.743, cos (0) = 0.557, cos (0%) = -0.371. A X B y Determine the distance between the satellites. (Round the final answer to four decimal places.) The distance between the satellites is km.

A longbowman launches arrows at 50 m/s. Assuming that the firing is done at 45 degrees from the horizontal and that there is no air resistance, what is the gained improvement in maximal range to ground targets when he fires from the skirt wall (height=10m), secondary wall (height=20m), and keep (height=30m) of a medieval castle? What if the castle is built on a hill of height 100m?

In a video game, airplanes move from left to right along the path described by y = 2 + 1 x. They can shoot rockets tangent to the direction of flight at targets on the x-axis located at x = 1 , 2 , 3 ,and 4. Will a rocket fired from ( 1 , 3 ) hit a target? Will a rocket fired from ( 2 , 2.5 ) hit a target? Will a rocket fired from ( 3 , 2.25 ) hit a target?