Special Cases
Exploiting ‘special’ (or ‘limiting’) cases is a powerful strategy, useful at all levels of physics. You met the idea in the section on Problem Solving Guidelines. You should remind yourself about it before proceding - it is Guideline 0.10.
To allow you to practice this strategy we give you three examples (progressively harder!) In each case we give you the result of a little bit of problem solving; we don’t ask you to reproduce the problem solving itself. We just want you to identify and check out the appropriate special cases of the ‘result’ provided.
Ants
A male and a female ant are separated by a distance D. They begin moving toward one another; the male ant moves at speed u1; the female at speed u2. When they meet the male ant has covered a distance
Balls
A ball of mass m1 travelling along a line with velocity u strikes a second ball of mass m2, which is travelling with the same speed in the oposite direction. The collision is head-on and elastic.
The velocity of the first ball after the collision is
Einstein
According to Einstein’s theory of relativity the kinetic energy of a body of mass m moving at speed v is
K=(γ(v)-1)mc2where
with c the speed of light.
Solution
Reveal

Solution
Ants
The obvious special cases are
The male is much faster than the female.
Then we’d expect d1 to be close to D.
Since
this checks out.The male is much slower.
Then we’d expect d1 to approach zero; it does.
- They move at equal speeds. Then we’d expect d1=D/2; it is.
Balls
Here the special cases are
- Mass m1 much bigger than m2. Then we’d expect v1 to be close to u (the incoming mass hardly notices the collision). It is.
- The masses are equal. Then we’d expect that both masses will recoil from the collision with the same speed, which must be u to conserve energy. The velocity of the first mass must then be then v1=-u. This checks out.
Mass m2 much bigger than m1. It is easy enough to read off the limiting case here too:
But it takes some more thought to see why this has to be so. An extra chocolate biscuit to whoever comes up with the simple explanation.
Einstein
The expression looks a bit fearsome. And it implies some very weird things.
But it also does behave sensibly in the limit in which we’d expect it to, namely when the speed v is small compared to c.
First imagine switching v off altogether then
and so the kinetic energy is zero for a body at rest.
This much is common sense.
The next bit is harder; and you may not know the maths yet (you will meet it this year).
If v is small compared to c, to an excellent approximation
and sowhich should feel like an old friend.