Two hot air balloons are in a race and are moving vertically, balloon A is direc

Question

Two hot air balloons are in a race and are moving vertically, balloon A is directly above balloon B as shown below. When the top of balloon B is 30 m above the ground and descending at 5 m/s, balloon A is ascending at 8 m/s. At this moment balloon A drops a spear to puncture balloon B so they can win the race. The spear is 15 m above the top of balloon B when it is released from rest, even with the bottom of balloon A. Each balloon is 6 m tall. Answer the questions below.

a) What is the maximum height above the ground reached by the spear?

b) How much time passes after the spear is released to when it hits balloon B?

c) How high is the basket of balloon B when the spear punctures it?

d) How high is the basket of balloon A the moment balloon B hits the ground? Assume balloon B is in freefall the moment it is punctured.

I have solved a and b successfully but I have attempted to solve c and d. I always get unreasonable answers.

Explanation / Answer

a)
When the spear is released it has 8m/s up velocity.

Hence,
v = u + at
0 = 8 -9.8*t
t = 0.816 sec

Maximum height of the spear is
s = ut +1/2*at^2
S = 8*0.816 -0.5*9.8*0.816^2
s = 3.265 m above the release point
Hence
from ground = 3.265 + 30 + 15 = 48.265 m

b)
Maximum Height from Baloon B of the spear will be at time 0.816 sec
So in 0.816 sec the baloon will travel 0.816*5 m vertically down.

Hence the distance between the spear and the baloon is 15+3.265+0.816*5 = 22.34 m

Now Spear has to cover this distance to hit the baloon.
Let us assume that the spear takes t time from the top to hit the baloon.

Baloon will travel = 5*t
And spear will travel : s = -1/2*9.8*t^2

Now since it hits the baloon
we can equate :
1/2*9.8*t^2 = 5*t + 22.34
9.8t^2 - 10t - 44.68 = 0
t = 2.71 sec

Total Time passed from release = 0.816 + 2.71 = 3.526 sec

c)
So in 3.526 the baloon will travel 3.526*5 vertically down
Hence distance from the ground = 30 - 3.526*5 = 12.37
Distance of basket = 12.37 - 6 = 6.37 m

d) Time taken by baloon b to hit ground
6.37 = 1/2*9.8*t^2
Hence t = 1.14 sec

Total Time = 3.526 + 1.14 = 4.66
Distance traveled = 8* 4.66 = 37.328 m

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