Von Neumann
- 101
- 4
Problem:
You see a jet plane flying and you think that it is flying at a constant altitude h. Say you hear the sonic boom at a time t after the plane passes directly overhead. Show that if the speed of sound v is the same at all altitudes, the speed of the plane is
v_s=hv/sqrt(h^2-v^2*T^2)
Partial Solution:
The sound travels a distance vT and the plane travels a distance v_s*T before the boom is found. Correct? However, I need to relate this to the height. If you assume h^2=(vT)^2+(v_s*T)^2. Since the speed of sound is the same at all altitude then the boom forms a perfect cone with angle θ with the ground. Then substituting in sinθ=v/v_s you can easily get the expression. However I don't fully understand the assumption h^2=(vT)^2+(v_s*T)^2 because geometrically it doesn't make sense to me. Any insight?
You see a jet plane flying and you think that it is flying at a constant altitude h. Say you hear the sonic boom at a time t after the plane passes directly overhead. Show that if the speed of sound v is the same at all altitudes, the speed of the plane is
v_s=hv/sqrt(h^2-v^2*T^2)
Partial Solution:
The sound travels a distance vT and the plane travels a distance v_s*T before the boom is found. Correct? However, I need to relate this to the height. If you assume h^2=(vT)^2+(v_s*T)^2. Since the speed of sound is the same at all altitude then the boom forms a perfect cone with angle θ with the ground. Then substituting in sinθ=v/v_s you can easily get the expression. However I don't fully understand the assumption h^2=(vT)^2+(v_s*T)^2 because geometrically it doesn't make sense to me. Any insight?