- #1

- 422

- 108

- Homework Statement
- A large mass of ##10^8## kg is held in free space at a certain point. A small mass of 1 kg exists at a point 1000 km (##10^6\;\text{m}##) away from the larger mass at the start of motion. Answer the following questions :

(1) Write the velocity of the smaller mass as function(s) of its space coordinate and time, assuming the location of the larger mass to be at the origin, the motion taking place along a line and the start of motion to be at ##t=0##.

(2) What is the location of the smaller mass at a given time ##t##?

(3) When will the smaller mass collide with the bigger mass?

(4) With what velocity will the bigger mass collide with the smaller mass?

- Relevant Equations
- From Newton's gravitation formula, the acceleration due to gravity ##g=\frac{d^2x}{dt^2}= \frac{GM}{x^2}##, where ##x## is the distance of separation between the masses ##M## and ##m## at a given instant.

**I draw a picture of the problem situation and paste it to the right.
Diagram : **

**Attempt :**Let me assume that the position of the smaller mass ##m## at a given instant of time ##t## is ##x(t)##.

(1) Gravitational potential energy ##\frac{GMm}{x} = \frac{1}{2}mv^2(x)##, where ##v(x)## is the velocity of the mass at that position. This simplifies to ##\boxed{v(x) = \sqrt{\frac{2GM}{x}}}##.

However, I do not know how to find the velocity as a function of the time ##t##, or ##v(t)##.

More crucially, I do not know how to solve the differential equation given in the Relevant Equations above : ##\frac{d^2x}{dt^2}= \frac{GM}{x^2}##. If I could, it would yield ##\frac{dx}{dt} = v(t)## and ##x(t)##, thereby answering questions 1 and 2 above.

Solving the differential equation would also lead to finding when will the masses collide and with what velocity, which would answers questions (3) and (4) above.

**Request :**A help or hint would be very welcome.