- #1
Benzoate
- 422
- 0
Homework Statement
In an air show, a pilot is to execute a circular loop at the speed of sound(340 m/s) . The pilot may black out if his acceleration exceeds 8g. Find the radius of the smallest circle he can use. [Take g=10 m s^-2
Homework Equations
possible equations: u^2 >= 2*M*G/a
m*dv/dt=-m*M*G/r^2
m*v^2/r=-m*M*G/r^2
The Attempt at a Solution
u=340 m/s
a=dv/dt=>=80 m s^-2
M=6.00e24 kg
G=6.67e-11
acceleration=dv/dt=> dv/dt=-M*G/r^2
option one
r=sqrt(M*G/(dv/dt)=7.07e12 meters
option 2
a=2*M*G/u^2, a being minimal radius and not the acceleration
a= 28284271.25 meters
actual answer: r>=1445 meters
what did I do wrong? should I have used polar coordinates since the problem states that the plane is going around a loop?
Perhaps I should write:
m*dv/dt=F(z) , F(z) representing the sum of all forces. the only two forces acting on the object is the gravitational force and the centripetal force ; so my equation looks like:
m*dv/dt=m*v^2/r-m*g
dv/dt=dv/dr*dr/dt=dv/dr*v. Now I can integrate in terms of the velocity vector and radius vector.
dv/dr=v^2/r-g
I get something that looks like this:
dv/v=dr/r-dr/g ==> ln v = In r -r/g
Last edited: