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