MATH 550

Professor Girardi

1997 Exam 1 Problem 5

Inspired by a most memorable call into Car Talk .

Houston, we have a problem. The space shuttle Atlantis is traveling along with position vector

r (t) = < t2 , 3t2 , 4t > .
If the power thrusters are turned off at time t, the Atlantis will coast off, with constant speed along a straight path tangent to the vector   r (t). The Atlantis is almost out of fuel when astronaut John Grunsfeld notices the Mir space station off ahead of them at the position Q = ( 220, 660, 64 ). John realizes that their only hope is to turn the thrusters off, just at the proper time, so that the Atlantis will safely coast to dock with the Mir; but, John is not sure if his plan will work. So John quickly calls Tom and Ray for advice. Tom claims that John's plan will work; Ray claims that John's plan will not work. Who is right: Tom or Ray? Why?

If astronaut Grunsfeld turns the power thrusters off at time t0 > 0 , then the Atlantis will coast off:
Way #1

Let v0 be the vector from the point P0 to the point Q , so

v0 = < 220 - t02 , 660 - 3t02 , 64 - 4t0 > .
Atlantis will hit Mir if and only if
  1. v0 is in the same direction of   r' (t0)
i.e. if and only if
  1. v0 is some positive multiple of   r' (t0)
i.e. if and only if
  1. for some positive number k ,
    v0 = k r' (t0) ,
i.e. if and only if
  1. for some positive number k ,
    220 - t02 = k 2 t0
    660 - 3t02 = k 6 t0
    64 - 4t0 =k 4 ,
i.e. if and only if
  1. for some positive number k ,
    t02 + 2 k t0 - 220 = 0
    t0 + k = 16 .
In 5, solving the latter equation for k and substituting into the former equation yields:
  1. t02 + 2 ( 16 - t0 ) t0 - 220 = 0 ,
which factors as:
  1. (t0 - 22 ) (t0 - 10 ) = 0 .
From this one can easily check that the solutions
  1. t0 = 10 and k = 6
    or
    t0 = 22 and k = -6
both satisfy the two equations in 5, but we need k to be positive so we will take the solution
t0 = 10 and k = 6 .
So if astronaut Grunsfeld cuts the power thrusters off at time t0 = 10, then the Atlantis will safely dock with MIR. Tom is correct.

Way #2

After having its power cut off at time t0, the Atlantis will travel along a line paratermized by

  1. l ( k ) = r (t0) + k r' (t0)
where 0 < k < infinity. So we want to find t0 and k so that
  1. 0 < t0 < infinity     and     0 < k < infinity     and
    r (t0) + k r' (t0) = <220, 660, 64 > ,
which is equivalent to
  1. 0 < t0 < infinity     and     0 < k < infinity     and
    <220, 660, 64 > - r (t0) = k r' (t0) .
Oh ...compare 3 to 11 ... they are the same. So we know that 11 also has a solution of t0 = 10 and k = 6 . So if astronaut Grunsfeld cuts the power thrusters off at time t0 = 10, then the Atlantis will safely dock with MIR. Tom is correct.

In case you are curious, the class vote was:
Abstentions 2
Tom was correct 10
Ray was correct 7
Both were correct 1
Neither were correct    0

In case you are bored, here's a copy ( dvi   ps   pdf) of a letter written to Tom and Ray.

Findable from URL: http://www.math.sc.edu/~girardi/