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Draw a line of equally-spaced dots, representing the locations of the aeroplane at some time interval. You can work back from the last dot (the current location of the 'plane), and draw circles out from each dot representing where the wavefront would be, from when the 'plane was at that point. So for the case when $v = c/2$, draw a circle of radius 2 from the 2nd-last dot, 3 from the 3rd-last dot, and so on.
Above the speed of sound, a mach cone arises around the aircraft. This can often be seen. If you draw out the diagram as above for the case when the speed of the 'plane is greater than the speed of sound, you can see from trig that the semi-apex angle of the cone is equal to $\sin^{-1}{(c/v)}$.
A solution to the $\sin{\theta} = 2$ problem is here.
Mach Attack
Age 16 to 18
Challenge Level 
Draw a line of equally-spaced dots, representing the locations of the aeroplane at some time interval. You can work back from the last dot (the current location of the 'plane), and draw circles out from each dot representing where the wavefront would be, from when the 'plane was at that point. So for the case when $v = c/2$, draw a circle of radius 2 from the 2nd-last dot, 3 from the 3rd-last dot, and so on.
Above the speed of sound, a mach cone arises around the aircraft. This can often be seen. If you draw out the diagram as above for the case when the speed of the 'plane is greater than the speed of sound, you can see from trig that the semi-apex angle of the cone is equal to $\sin^{-1}{(c/v)}$.
A solution to the $\sin{\theta} = 2$ problem is here.
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