Question

# In the Hunger Games, Katniss shoots an arrow at an apple in a pig’s mouth using...

In the Hunger Games, Katniss shoots an arrow at an apple in a pig’s mouth using a longbow. After careful analysis of the scene, my daughter and I estimate she is standing 12 m away and 3.5 m below her target. A typical shot from a longbow reaches speeds of 55 m/s.

a) At what angle above the horizontal must she fire the arrow? There are two solutions (what do they correspond to?) and which one would be the one she used (she wanted a sudden dramatic impact). Treat the arrow as a simple projectile.

b) How fast is arrow moving when it hits the apple?

c) After hitting the apple the arrow takes the apple with it and hits the wall half a meter behind the pig, how fast are the arrow and apple moving as they hit the wall? (A typical arrow has a mass of 18 grams, and a typical apple has a mass of 100 grams).

d) Suppose we model the bowstring as an ideal spring with spring constant k= 70 N/m, how far must Katniss draw the bow to give it the arrow an initial velocity of 55 m/s?

(A) x= horizontal distance of apple

y= vertical distance of apple

using y= x tan - gx2/2u2cos2

3.5 = 12 tan - 10×12×12/2(55×55)(cos2)

3.5 = 12 tan - 0.238sec2

3.5 = 12 tan - 0.238( 1 + tan2)

0.238 tan2 - 12 tan + 3.738=0

tan= (-(-12)+-(122-4×0.238×3.738)0.5)/(2×0.238)

= 17.400 or 72.600 with horizontal

She would shoot the arrow at 17.40 degree with horizontal for sudden impact.

(B) let speed of arrow on hitting apple =V

V=( (u cos)2 + ( u sin - gt)2)1/2

Where t= time taken to hit apple

t= x/u cos = 12/ 55 cos(17.40)= 0.23 sec

V= ((55 cos(17.40))2 +(55 sin 17.40 - 10×0.23)2)1/2

V= 54.36 m/s

(D) gain in kinetic energy by arrow = loss in potential energy of spring, let L = length drawn by arrow

(0.5 mu2) = (0.5 kL2)

L= (mu2/k)1/2= ( 0.018×55×55/70)1/2 = 0.881 m

L= 88.1 cm

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