0

Trying to find the analytical equation for instantaneous velocity of an object under the influence of gravity (non-uniform acceleration) in a straight line. I am trying to model a stationary massive object (ex. sun) of mass M attracting a small ball of mass m and trying to find the small object's velocity and position. I know the force would be given by:

$$F = \frac{G M m}{d^2}$$

and that the acceleration of small object is $F/m$. I also know that the integral of acceleration with respect to time $t$ equals the velocity. So does that mean the equation for velocity would be:

$\large{v(t) = v_0 + \frac{GM}{d^2}t}$

and position would be

$\large{d(t) = d_0 + v_0t + \frac{1}{2}\frac{GM}{d^2}t}$

that doesn't seem right. I know since its non-uniform acceleration its not that simple but i don't know how to solve it. Any help would be greatly appreciated.

UKH
  • 5,021
user3712173
  • 3

1 Answers1

3

No, your result is incorrect. Remember that $d$ is a function of $t$, so $\frac{G M}{d^2}$ is not a constant and your result for $v(T)$ (and consequently $d(t)$, for the same reason) is wrong. You need to solve the differential equation $\ddot d(t)=-\frac{G M}{d(t)^2}$.

Pirx
  • 3,803