Suppose that you are moving along the –axis and that at time your position is given by
We’re going to try and get a good picture of what your motion is like. We can learn quite a bit just by looking at the sign of the velocity at each time
- If
then at that instant is increasing, i.e. you are moving to the right. - If
then at that instant you are not moving at all. - If
then at that instant is decreasing, i.e. you are moving to the left.
From the given formula for it is straight forward to work out the velocity
This is zero only when and when at no other value of can this polynomial be equal zero. Consequently in any time interval that does not include either or takes only a single sign . So
1
This is because the equation is only satisfied for real numbers and when either or or both Hence if a polynomial is the product of two (or more) factors, then it is only zero when at least one of those factors is zero. There are more complicated mathematical environments in which you have what are called “zero divisors” but they are beyond the scope of this course.
2
This is because if and then, by the intermediate value theorem, the continuous function must take the value for some between and
- For all
both and are negative (sub in, for example, ) so the product - For all
the factor and the factor (sub in, for example, ) so the product - For all
both and are positive (sub in, for example, ) so the product
The figure below gives a summary of the sign information we have about and
It is now easy to put together a mental image of your trajectory.
- For
large and negative (i.e. far in the past), is large and negative and is large and positive. For example, when3
Notice here we are using the fact that when is very large is much bigger than and So we can approximate the value of the polynomial by the largest term — in this case We can do similarly with — the largest term is and So you are moving quickly to the right. - For
so that is increasing and you are moving to the right. - At
and you have come to a halt at position - For
so that is decreasing and you are moving to the left. - At
and you have again come to a halt, but now at position - For
so that is increasing and you are again moving to the right. - For
large and positive (i.e. in the far future), is large and positive and is large and positive. For example, when4
We are making a similar rough approximation here. and So you are moving quickly to the right.
Here is a sketch of the graphs of and The heavy lines in the graphs indicate when you are moving to the right — that is where is positive.
And here is a schematic picture of the whole trajectory.