The last two examples, in sections 3.3.5 and 3.3.4 posed a different kind of problem then we considered before. We started off in section 3.3.2 by asking
, 
, and t, what is x?
The last two examples asked the question
, , and v, what is x?
Which is an equally legitimate question. Do we want to
go through the same rigmarole as in the last example?
It'd be better to find the general formula relating
the above quantities a, , , v, and x.
We almost had that formula but we blew it, I'm
sorry to say. Yes, I'm going to get preachy
again and mention yet another rule that you should
always follow, (well maybe not always, but most of the time).
Do not substitute numerical values into formulas until you've got to the last stage of the problem. Do everything in terms of symbols, not numbers.
I'll illustrate that for you with the problem at hand.
Eq. 3.17 gave the general expression relating
t to v, and a. We screwed up getting a
nice general formula by foolishly substituting in for
the numerical values of quantities, giving us 
.
That wasn't necessary at all. We could have waited
to do the substitution. Let's see what happens if
we do that now.
Following the same logic we used in that example, the next thing to do was to substitute t into eq. 3.14. Instead of substituting in 1s let's substitute what's after the second equals sign in eq. 3.17, and keep all other quantities as symbols.
This can easily be simplified. It's just a little algebra. You finally get
There we are. A general formula relating
a, , , v, and x! That's what we wanted.
With this formula, you can solve a lot of problems similar
to examples 3.3.4 and 3.3.5
quite simply.
Another reason why doing things in symbols is so
important is that it allows you to check you answers
to see if they make sense. Look at eq. 3.20.
Check what happens if a=0. Well then you see that
the speed of the particle doesn't change. That sounds
correct. How about if 
? Yes the initial and final speeds
are the same in that case also. You know in these
limits, the answer is right. Also you can easily
check that the units work out. This gives you some
confidence that the formula is indeed correct.
If you had instead plugged in the time as we did
in the previous example (3.3.5), you
couldn't check all these limits and would be
far more likely to make some mistake.
It takes a while to get used to solving problems with symbols instead of numbers. It seems far too abstract for most people at first. You can get over this problem by using the same approach we did here. First solve the problem the more comfortable way by plugging in numbers as you go along. But after that go back, and go through the same steps but this time keep all your symbols. It's the same logic in both cases. You're just replacing a lot of multiplication and addition with algebraic manipulations.
