## Poll

 I love math! 16 votes (50%) Math is great. 12 votes (37.5%) My religion is mathology. 5 votes (15.62%) Women didn't speak to me until I was 30. 2 votes (6.25%) Total eclipse reminder -- 04/08/2024 10 votes (31.25%) I steal cutlery from restaurants. 3 votes (9.37%) I should just say what's on my mind. 6 votes (18.75%) Who makes up these awful names for pandas? 5 votes (15.62%) I like to touch my face. 10 votes (31.25%) Pork chops and apple sauce. 7 votes (21.87%)

32 members have voted

chevy
Joined: Apr 15, 2011
• Posts: 86
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December 14th, 2020 at 8:12:13 AM permalink

Volume of hemisphere = volume of cone
(2/3) * pi * r^3 = (1/3) * pi * r^2 * h
h=2*r

h=8cm
Gialmere
Joined: Nov 26, 2018
• Posts: 2057
December 14th, 2020 at 3:13:33 PM permalink
Quote: rsactuary

h=8

using volume of a hemisphere = 4/6 x pi x r^3

and volume of a cone = pi x r^2 X h/3

Quote: chevy

Volume of hemisphere = volume of cone
(2/3) * pi * r^3 = (1/3) * pi * r^2 * h
h=2*r

h=8cm

Correct!
---------------------------

Today I realized that eating ice cream isn't filling the emptiness I feel inside.

But I'm no quitter.
Have you tried 22 tonight? I said 22.
Gialmere
Joined: Nov 26, 2018
• Posts: 2057
December 15th, 2020 at 8:07:23 AM permalink
It's toughie Tuesday. Time for rocket science...

You are a pyrotechnician in charge of the nightly firework display at an amusement park. You've received some new style rockets from Europe and are testing one in order to time it to your show's music soundtrack.

The firework rocket is fired vertically upwards with a constant acceleration of 4 ms^-2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 meters where it detonates.

Assuming no air resistance and taking g=9.8 ms^-2, how long does it take the rocket to reach its maximum height?
Have you tried 22 tonight? I said 22.
Wizard

Joined: Oct 14, 2009
• Posts: 23328
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December 15th, 2020 at 9:46:26 AM permalink
Quote: Gialmere

Assuming no air resistance and taking g=9.8 ms^-2, how long does it take the rocket to reach its maximum height?

Apx. 9.85 seconds.

Not that you asked, but the rocket fuel burns out in seven seconds.

I'll provide a solution (hopefully) if my answer is right
It's not whether you win or lose; it's whether or not you had a good bet.
ChesterDog
Joined: Jul 26, 2010
• Posts: 965
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December 15th, 2020 at 10:37:13 AM permalink
Quote: Gialmere

It's toughie Tuesday. Time for rocket science... ...
You are a pyrotechnician in charge of the nightly firework display at an amusement park. You've received some new style rockets from Europe and are testing one in order to time it to your show's music soundtrack.

The firework rocket is fired vertically upwards with a constant acceleration of 4 ms^-2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 meters where it detonates.

Assuming no air resistance and taking g=9.8 ms^-2, how long does it take the rocket to reach its maximum height?

I agree with the Wizard's answer.

Expressed algebraically, I get a total time of [ 2H ( 1/a + 1/g ) ]1/2, where a = 4 meters per sec2 and H =138 meters.

ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4930
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December 15th, 2020 at 3:02:50 PM permalink

Let a be the time the fuel burns
At time a, the distance = 1/2 x 4 x a^2 = 2 a^2
and the velocity = 4 a

Let b be the time from when the fuel runs out to when the rocket explodes
At time b, the distance = 4 ab + 1/2 x (-9.8) x b^2 = 4 ab - 4.9 b^2
and the velocity = 4 a - 9.8 b = 0
a = 2.45 b

The total distance = 2 a^2 + 4 ab - 4.9 b^2 = 138
2 (2.45 b)^2 + 9.8 b^2 - 4.9 b^2 = 138
(4.9 x 2.45 + 4 x 2.45 - 2 x 2.45) b^2 = 138
b^2 = 138 / (6.9 x 2.45)
b = sqrt(138 / (6.9 x 2.45))
The total distance = a + b = 3.45 sqrt(138 / (6.9 x 2.45)) = 69 / 7 m

ssho88
Joined: Oct 16, 2011
• Posts: 552
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December 15th, 2020 at 4:01:11 PM permalink
Quote: Gialmere

It's toughie Tuesday. Time for rocket science...

You are a pyrotechnician in charge of the nightly firework display at an amusement park. You've received some new style rockets from Europe and are testing one in order to time it to your show's music soundtrack.

The firework rocket is fired vertically upwards with a constant acceleration of 4 ms^-2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 meters where it detonates.

Assuming no air resistance and taking g=9.8 ms^-2, how long does it take the rocket to reach its maximum height?

g=9.8, a = 4, H = 138

t1 =( 2gH/(ag+a^2))^0.5 = 7

t2 = (2aH/(g^2+ag))^0.5 = (1104/135.24)^0.5 = 2.857

Total time = 9.857 sec

S1 = 1/2a(t1)^2 = 98

S2 = u(t2) - 1/2g(t2)^2 = 4*7*(1104/135.24)^0.5 - 4.9*(1104/135.24) = 40

Gialmere
Joined: Nov 26, 2018
• Posts: 2057
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December 15th, 2020 at 5:05:05 PM permalink
Quote: Wizard

Apx. 9.85 seconds.

Not that you asked, but the rocket fuel burns out in seven seconds.

I'll provide a solution (hopefully) if my answer is right

Quote: ChesterDog

I agree with the Wizard's answer.

Expressed algebraically, I get a total time of [ 2H ( 1/a + 1/g ) ]1/2, where a = 4 meters per sec2 and H =138 meters.

Quote: ThatDonGuy

Let a be the time the fuel burns
At time a, the distance = 1/2 x 4 x a^2 = 2 a^2
and the velocity = 4 a

Let b be the time from when the fuel runs out to when the rocket explodes
At time b, the distance = 4 ab + 1/2 x (-9.8) x b^2 = 4 ab - 4.9 b^2
and the velocity = 4 a - 9.8 b = 0
a = 2.45 b

The total distance = 2 a^2 + 4 ab - 4.9 b^2 = 138
2 (2.45 b)^2 + 9.8 b^2 - 4.9 b^2 = 138
(4.9 x 2.45 + 4 x 2.45 - 2 x 2.45) b^2 = 138
b^2 = 138 / (6.9 x 2.45)
b = sqrt(138 / (6.9 x 2.45))
The total distance = a + b = 3.45 sqrt(138 / (6.9 x 2.45)) = 69 / 7 m

Quote: ssho88

g=9.8, a = 4, H = 138

t1 =( 2gH/(ag+a^2))^0.5 = 7

t2 = (2aH/(g^2+ag))^0.5 = (1104/135.24)^0.5 = 2.857

Total time = 9.857 sec

S1 = 1/2a(t1)^2 = 98

S2 = u(t2) - 1/2g(t2)^2 = 4*7*(1104/135.24)^0.5 - 4.9*(1104/135.24) = 40

Correct!

Very good.

Wow! We have quite a few rocket scientists here. How are you guys at brain surgery?
---------------------------------

Here is the pyrotechnician's official mug...

And...

[This is probably my last puzzle until after the new year so, Happy Holidays math gamers!]
Have you tried 22 tonight? I said 22.
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4930
December 15th, 2020 at 7:17:10 PM permalink
Quote: Gialmere

Wow! We have quite a few rocket scientists here.

Actually, real rocket scientists would have pointed out that gravity is not constant, but varies as the inverse square of the distance to the center of gravity, and you also have to take into account the decrease in the rocket's mass as the fuel is used up.
ssho88
Joined: Oct 16, 2011
• Posts: 552
December 15th, 2020 at 8:33:07 PM permalink
Quote: ThatDonGuy

Actually, real rocket scientists would have pointed out that gravity is not constant, but varies as the inverse square of the distance to the center of gravity, and you also have to take into account the decrease in the rocket's mass as the fuel is used up.

Air resistance/surface resistance should decrease when rocket go higher, will surface resistance decrease when surface temperature increase ? The Coriolis Effect ?

For knowledge purposes only.
Last edited by: ssho88 on Dec 15, 2020