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A hot air balloon is rising straight upward with a constant speed of 8.80 m/s. When the...

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brandih | eNotes Employee

Posted October 7, 2013 at 1:54 AM via web

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A hot air balloon is rising straight upward with a constant speed of 8.80 m/s. When the basket of the balloon is 100.0 m about the ground a bag of sand tied to the basket comes loose. What is the greatest height of the bag of sand during its fall to the ground?

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llltkl | College Teacher | (Level 3) Valedictorian

Posted October 7, 2013 at 2:33 AM (Answer #1)

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When the bag of sand comes loose from the basket of the balloon, it will have two forces acting on it in mutually opposite directions: one upward due to the upward velocity of the balloon and the other downward acceleration due to gravity. As a consequence, it will rise a bit and then start falling to the ground.

For the bag of sand,
Initial upward velocity`=u`
final velocity, `v = 0` (since at the topmost point its upward velocity will become zero)
`g= 9.81` m/s^2
Applying
`v=u-g*t` (since g is downwards)
we get time to rise up, `t=(u/g)` s

Height attained in this time, in addition to the initial height of 100 m, can be obtained by applying

`s = u*(u/g)-1/2 g(u/g)^2`

`=u^2/(g) -1/2*g*(u/g)^2`

`=u^2/(g)-u^2/(2g)`

`=u^2/(2g)`

`=8.8^2/(2*9.81)` m

`=3.95` m

So, total height of the bag of sand from the ground before coming to a standstill and then beginning to fall:

`= (100+ 3.95)` m

`=103.95` m

This is the greatest height of the bag of sand during its to fall to the ground.

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted October 7, 2013 at 3:41 AM (Answer #2)

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  • When the balloon goes upward with a velocity of 8.8m/s all items inside it has the same velocity of 8.8m/s.
  • When the sand bag get loosed it also have that speed at the moment of loose.
  • The balloon goes upwards because it has a force upward because of hot air.
  • So all the parts inside the balloon is subjected to this force.
  • When the sand bag get loosened it will loss the connection with the balloon.
  • So this force due to hot air will no longer apply on sand bag.
  • Due to the sand bag's weight it should come down.
  • Since it has a velocity of 8.8m/s at the moment of loose it will go some distance upward under gravity and then start to come down.

Using equation of motion;

`uarrv^2 = u^2+2as`

`v = 0` since at the greatest height it suddenly get to rest.

`u = 8.8m/s`

`g = -9.8m/s^2` Since the bag travels under gravity

`s = ?`

`uarrv^2 = u^2+2as`

`0 = 8.8^2-2xx9.81xxs`

`s = 3.95m`

So the height from ground is `100+3.95 = 103.95m.`

Assumption

There is no air resistance applied on the sand bag.

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