Computer Algebra Recipes

200 CHAPTER 4. NONLINEAR ODE MODELS
OK, X, there's your curve shown in Figure 4.17. You can arrange for a window
to be left open at the point where the curve intersects the museum roof.
Now that we know the distance through which you will be dropping, we
can easily calculate your velocity from the expression v = p 2 g y1. Takingthe
gravitational acceleration to be g =9:81 m/s 2 ,
> g:=9.81: vel:=sqrt(2*g*y1);
vel := 49:98095625
you will be traveling at almost 50 m/s as you pass through the window on
the museum roof, if you haven't been braking. This is an upper bound on
your speed, of course, because we have completely neglected friction and air
resistance. I can see by the look on your face that you don't quite appreciate
how fast this is. So, let me convert the velocity to kilometers per hour.
> vel2:=convert(vel,units,m/s,km/h)*km/h;
179:9314425 km
vel2 :=
h
Your theoretical speed would be almost 180 km/hr but, as I said, in reality it
would be somewhat less. The time of descent can now be calculated from
Z £
T =
p
(dx=dμ) 2 +(dy=dμ) 2
dμ;
2 gy(μ)
viz:;
μ=0
> T:=int(sqrt(diff(x,theta)^2+diff(rhs(Y),theta)^2)
/sqrt(2*g*rhs(Y)),theta=0..evalf(Theta));
T := 5:659009626
The minimum time of descent is about 5:7 seconds. Again, friction, air resistance,
and any braking on your part would lengthen this time. Given the
estimates that we have come up with, do you still want to go through with your
bizarreescaperoute?"
\Sure, Mike," Mr. X replies, \and I am counting on you to help attach the
wire to the appropriate window in the museum roof and leave it open for me."
\Oh no," Mike groans. \If I get caught, Vectoria's parents are sure to call
o® our recently announced engagement."
PROBLEMS:
Problem 4-41: A di®erent escape route
Determine Mr. X's escape route if the museum roof slanted at 30 ± to the horizontal,
the street is 50 m wide, and the vertical section of the museum wall
adjoining the slanted roof is 25 m tall. Make a plot similar to that in Figure
4.17. How close to the edge of the roof would Mr. X land? What would be
his speed at this point? Neglect friction and air resistance.
Problem 4-42: In search of reality
Suppose that Mr. X decides that sliding down a steel wire at nearly 180 km/hr
is a little too insane even for his bizarre tastes. His friend Mike is unavailable for
technical advice because he is busy with preparations for his upcoming wedding