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- November 20th 2006, 08:37 PM #1

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## Question 7

I hope this is an easy one

A thin semi-infinite rod is made of high explosive. The detonation speed is v

(that is the speed at which the detonation propagates down the rod is v)

The speed of the detonation products through space is u<v.

Describe the shape of the shape of the explosion in time and space. That is

what is the time evolution of the region occupied by the explosion products.

RonL

- November 21st 2006, 02:50 AM #2

- November 21st 2006, 03:17 AM #3

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- November 21st 2006, 03:48 AM #4

- November 21st 2006, 08:02 AM #5

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Would it be a solid cone that increases in base radius at the speed of the explosion products and hiegth at the speed of the explosion's movment up the rod? At the base of the cone, there is a semisphere whose radius increases with the explosion products' speed as well.

The cone angle is 90u / v˚, since slope can be found by rise (u) over run (v) and be compared to the angle of the x- and y-axes (the y-axis is the cone's hypotenuse). It will always be less than 90˚, since the speed of the explosion particles is less than that of the explosion itself.

- November 21st 2006, 12:53 PM #6

- November 21st 2006, 12:56 PM #7

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- November 21st 2006, 01:34 PM #8

- November 21st 2006, 02:19 PM #9

- November 26th 2006, 09:02 PM #10

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## Solution

First ask what happens when a single, tiny particle explodes. It will explode in all directions, giving a spherical explosion. The radius of the sphere containing the explosion products will expand at speed u.

Now fir the semi-infinite rod, consider it to be made up chain of particles of the explosive. Each particle generates a spherical explosion that spreads through space with speed u. However, the particles do not all explode at the same time. The explosion propagates down the rod with speed v. The result is a series of spheres, the largest growing from the end of the rod and having radius ut, and as you proceed down the rod to later explosions, successively smaller spheres, until you arrive at the point where the explosion is just occurring. The successively shrinking spheres add up, as shown in the attached figure, to a cone of altitude vt with a hemispherical cap of radius ut fitted perfectly on to the base of the cone.

RonL

- November 27th 2006, 11:49 AM #11

- November 27th 2006, 12:18 PM #12
I agree. Too bad MathGuru is considering dropping the program from the forum

To get that diagram, type in:

[draw]FixedPoint (130,50); FixedPoint (130,250); Ray (2,1) [hidden]; Point on object (3,.05) [green]; Point on object (3,0) [hidden]; Point on object (3,.9) [hidden]; Point on object (3,.8) [hidden]; Point on object (3,.7) [hidden]; Point on object (3,.6) [hidden]; Point on object (3,.5) [hidden]; Point on object (3,.4) [hidden]; Point on object (3,.3) [hidden]; Point on object (3,.2) [hidden]; Point on object (3,.1) [hidden]; Point on object (3,1) [hidden]; Segment (1,2) [thick]; Segment (1,4)[hidden]; Length (17,0,0,'d') [hidden]; FixedPoint (20,150) [label ('S')]; FixedPoint (20,250) [label ('V')]; Segment (19,20); Point on object (21,.5) [label ('U')]; Segment (19,22); Length (21,0,0,'d') [hidden]; Length (23,0,0,'25') [hidden]; Calculate (0,0,'26','A B /') (25,24) [hidden]; Calculate (0,0,'0','A 20 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 40 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 60 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 80 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 100 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 120 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 140 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 160 - B *') (18,26) [hidden]; Calculate (0,0,'0','A 180 - B *') (18,26) [hidden]; Calculate (0,0,'36','A 200 - B *') (18,26) [hidden]; Calculate (0,0,'37','A B *') (18,26); Circle by radius (1,37) [hidden]; Circle by radius (2,36) [hidden]; Circle by radius (6,35) [hidden]; Circle by radius (7,34) [hidden]; Circle by radius (8,33) [hidden]; Circle by radius (9,32) [hidden]; Circle by radius (10,31) [hidden]; Circle by radius (11,30) [hidden]; Circle by radius (12,29) [hidden]; Circle by radius (13,28) [hidden]; Circle by radius (14,27) [hidden]; Circle interior (38) [red]; Circle interior (39) [red]; Circle interior (40) [red]; Circle interior (41) [red]; Circle interior (42) [red]; Circle interior (43) [red]; Circle interior (44) [red]; Circle interior (45) [red]; Circle interior (46) [red]; Circle interior (47) [red]; Circle interior (48) [red];[/draw]

- November 27th 2006, 07:38 PM #13

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