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

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Giordano Bruno

Is there any specific direction in which detonation produvts would travel?

Keep Smiling
Malay

Unlike in SF movies the products from a small amount of explosive will expand
in a shperical shell.

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

what would we the centre of the spherical shell?
Thanks for your response

Keep Smiling
Malay

The point of detonation of the small quantity of explosive.

RonL

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.

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"I am empowered to lie. But I have never lied to you." - The Giver

Move the green point to see the explosion









The speed of the detonation products through space is set by the distance from S to U (which you can change)

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Quick's Prime Property:
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And what is the angle involved?

RonL

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Giordano Bruno

What angle?

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The angle the "cone" makes when projected against a flat surface. ie. what is the angle the top of the cone makes?

-Dan

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"I must not fear. Fear is the mind killer. Fear is the little death that brings total obliteration. I will face my fear. I will permit it to pass over me and through me. And when it has gone I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain." - The Litany Against Fear, "Dune" by Frank Herbert

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

Attached Thumbnails

 

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

A truly great illustration. That's pedagogy.
It reminds me of a sound barrier from a plane going in over the speed of sound.
And a really nice illustration from Quick as well.

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]

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It should do, its supposed to illustrate the formation of a Mach cone.

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno