Cacl 3 example problem

Question

Consider a spaceship that is traveling from the surface of the Earth to space

in a straight line. We are going to compute the work necessary to perform to make the spaceship can exit the

gravitational field of Earth (with some simplifications :-) ).

The (normalised) gravitational force is given by:

F(x, y, z) = 1/((x² + y² + z²)^3/2 )*(x, y, z).

Let us parametrise the trajectory of the spaceship in the following way: r(t) = t(1, 1, 1) for 1 ≤ t ≤ a, for

some a.

(a) Find the work done in moving the spaceship along the defined trajectory

(b) The spaceship escapes the gravitational field of Earth at a = ∞. What happens to the work if you let a → ∞?

(c) Consider a new trajectory given by: r(t) = t(2, 1, 1) for 1 ≤ t ≤ a, for some a. Is this a more efficient

trajectory

Answer 1

a) W(a) = ∫₁^aF•dr = ∫₁^adt/(√3t³) = - √3/(2t²)₁^a = √3/2(1 - 1/a²)

b) lim W(a) a →∞ = √3/2;

c) W = ∫₁^a4dt/(6√6t²) = -1/(3√6t²)₁^a = √6/18(1 - 1/a²) < √3/2(1 - 1/a²)

So, this way is more efficient