34 
The N.Z. Journal of Science and Technology. 
[Feb. 
Rectilinear Motion, 
I. Let a body initially at rest with respect to the Sun be drawn towards 
it by gravitation. It is acted on by an attractive force which varies 
inversely as the square of the distance, x. The equation of motion is, 
therefore, 
d 2 x /x 
72 > 
dt l 
X‘ 
where /a is the attractive force at unit distance, a constant to be determined 
later. 
Integrating, we have 
dx> 2 
dt. 
a 
where a represents the initial distance. 
If a is infinitely great, 
'dx\ 2 2 /x 
.dt) x 
or 
fX 
V* — - 
X 
1 - 1,2 
where v is the velocity at a distance x : that is, the kinetic energy of a 
body per unit mass when it arrives at a point at a distance x from the 
centre of attraction is 
[X 
X * 
The velocity at this point is given by the equation 
v = a/A. 
x 
Inverting each side of the velocity equation, we have 
dt x 
dx 2/x 
2 1 3 
' '\ ,v : 
where c is a constant. 
But since t vanishes when x does so, c = 0; 
.*. t = 
a/2 
Before drawing curves to scale to represent these equations we must 
determine the value of p. In order to do so we must first find G, the 
constant of gravitation, in the equation 
Attractive force = 
GMm 
I 2 " 
where M and m are the masses of two bodies and d the distance between 
them. 
