Darlene is a stunt motorcyclist in a traveling circus. For the climax of her show, she takes off from the ramp at angle θ, clears a fiery ditch of width x, and lands on an elevated platform (height H) on the other side (Figure). Darlene notices, however, that night after night, the circus owner keeps raising the height of the platform and the flames to make the jump more spectacular. She is beginning to worry about how far this trend can be taken before she becomes a spectacular casualty, so she decides that it is time for some calculations. 

(a) For a given angle θ and distance x, what is the upper limit H_max such that the bike can make the jump? 

(b) For H less than H_max, what is the minimum takeoff speed necessary for a successful jump? (Neglect the size of the bike.)

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