- #1
weirdoguy
- 1,066
- 1,010
Hi everyone. My high-school student got the following homework exercise at school which I have problem with:
Smoke from the steam locomotive is carried by horizontally blowing wind. Shape of the smoke trail is shown in the attached figure. Using the drawing, determine velocity of the wind assuming it is constant. Speed of the steam engine is ##v=60\frac{km}{h}## and it is moving on an arc of a circle.
My problem is that I really don't see any way to approach it on an elementary level. If there were no wind the smoke trail would be an involute of circle. I derived parametric equation of the trail for radius ##R=1m## and speed of the train ##v=1\frac{m}{s}## with ##\vec{v}_{initial}=[0,1]##, ##\vec{r}_{initial}=[1,0]## and ##\vec{v}_{wind}=[v_{xw},v_{yw}]##:
##x(t)=\cos t+(T-t)(v_{xw}-\sin t)##
##y(t)=\sin t +(T-t)(v_{yw}+\cos t)##
for ##t\in\langle 0,T\rangle##.
Playing with these a little bit I've noticed that the cusp seen in the attached picture appears only for quite specific range of values of ##v_{xw}## and ##v_{yw}##. For others there is a loop, and sometimes none of these. Anyway, it's still not clear for me how to approach this problem on a high-school level
Any thoughts?
Smoke from the steam locomotive is carried by horizontally blowing wind. Shape of the smoke trail is shown in the attached figure. Using the drawing, determine velocity of the wind assuming it is constant. Speed of the steam engine is ##v=60\frac{km}{h}## and it is moving on an arc of a circle.
My problem is that I really don't see any way to approach it on an elementary level. If there were no wind the smoke trail would be an involute of circle. I derived parametric equation of the trail for radius ##R=1m## and speed of the train ##v=1\frac{m}{s}## with ##\vec{v}_{initial}=[0,1]##, ##\vec{r}_{initial}=[1,0]## and ##\vec{v}_{wind}=[v_{xw},v_{yw}]##:
##x(t)=\cos t+(T-t)(v_{xw}-\sin t)##
##y(t)=\sin t +(T-t)(v_{yw}+\cos t)##
for ##t\in\langle 0,T\rangle##.
Playing with these a little bit I've noticed that the cusp seen in the attached picture appears only for quite specific range of values of ##v_{xw}## and ##v_{yw}##. For others there is a loop, and sometimes none of these. Anyway, it's still not clear for me how to approach this problem on a high-school level
Any thoughts?