TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Motion with Variable Mass
These notes are based on the requirements of the M5 A Level mathematics module.
Motion with variable mass
You only need to know the impulse-momentum principle:
.
From this you can set up a differential equation that describes the motion of the system.
e.g. A particle is moving upward against gravity and is losing mass at a rate k mass/sec. The lost mass is ejected vertically downwards with speed u relative to the body.
To start off, the particle is moving against gravity. So the force acting on it is its weight. Thus:
, where 
is the time interval through which this impulse occurs and δm is the increase in mass (see below).
Now we want to find the change in momentum.
(i) particle
, supposing that the body gains 
of mass and 
of speed (the fact that it loses mass will be accounted for later)
(ii) ejected mass
, we always ignore 
since it's small.
So the change in momentum of the whole system is:
Hence by the impulse-momentum principle:
Some rearrangement (remember, we ignore 
):
Divide by and take limits:
This is the differential equation that describes this motion. Now we take into account one last piece of information:
The body loses mass at constant rate, i.e. 
.
This transforms the differential equation into:
, which you can now solve easily.
Some things you should know when solving questions related to this chapter:
- Limiting speed is achieved when
.
- When dealing with rocket problems, sometimes the time of burnout helps. e.g. suppose that a rocket has variable mass m and that the mass of the fuel is Mf and of the rocket is Mr. The time of burnout T is when
. This fact is simple (and obvious), but easily over-looked.
- When dealing with rain-drop type problems, use mass=density*volume, i.e.
. This is helpful because sometimes you're told that the rain-drop is spherical, and this helps you get enough information to find an expression for 
. That's because you know that p is constant, 
, and you're usually given that 
is constant. In other words: 
.
Comments
Originally written by dvs on TSR forums.
