A space graft leaves the orbit of a planet circling the star Tau Ceti at a velocity of 10,000 miles/hr at an angle to the orbital tangent of 12 degrees. Immediately after departing, the space craft's engine cease and the craft is pulled toward the star (mass =

kg). Compute the craft's distance from the star as a function of time (assuming the engines are not repaired).

Specify a radial coordinate axis system with the radius toward the star as negative and away from the star as positive

Forces involved

The force due to gravity

The force required to accelerate the craft radially toward or away from the star is given by:

Since the force to accelerate the craft toward the star is just gravity, the following differential equation results:

or

or after simplifying

This is a nonlinear, 2nd order, homogeneous differential equation which we could easily solve by hand too.

Let's use Mathcad's Runge-Kutta [rkfixed function] to solve this problem:

The mass of Tau Ceti:

Universal Gravitational Constant:

remembering:

[G is negative because of our definition of the coordinate system]

specifying the initial conditions:

initial displacement

in an array named init

(meters)

initial velocity

(meters/sec radially outward)

define the first derivative

defining an array D showing the relationship of the derivatives (this works for higher order equations also).

the 2nd derivative (defined in terms of the 1st derivative)

specifying the interval where the solution is to be found using evaluated at a given number of points

defining R with the Mathcad function:

The solution shows an impact at about 117.5 hours.