For this part of the lab, load the worksheet

/usr7/bfarr/Maple/solve.mswhich contains one of the partial solutions to the problem from the previous lab. Go through the worksheet and execute all of the commands, ending with the plot. Notice that the command for finding the time

> t_hit := solve(y(t) = 5,t);

Next, type in the the following command on the first blank input
line. The notation `t_hit[2]`

is used to select the second
solution for `t_hit`

from the two shown above.

> solve(x(t_hit[2]) = 30,theta);

This first method has given us the two solutions we could see in the
plot. We could call this first method a sequential method, because we
first solved the equation for *y* and then the equation for *x*. An
alternative method that requires less typing is to solve the two
equations simultaneously with the following command.

> solve({x(t) = 30, y(t) = 5},{t,theta});

The first argument to the `solve` command always consists of the
equation(s) to be solved.
When you want to solve two or more equations simultaneously, you have
to enclose them in curly braces, {}. Otherwise, Maple would have
trouble figuring out how many equations there are. The second
argument to the `solve` command is the variable(s) to solve
for. The general rule is that you need as many variables to solve for
as you have equations. If you don't specify the correct number of
variables to be solved for, then Maple probably won't be able to solve
the equations. As was done for the equations, you must enclose the
variables to be solved for inside curly braces.

Maple uses curly braces to denote a set. In a set, order isn't
important so your output may have the solutions reported in a
different order. The way Maple orders the items when printing out a
set is done by memory location, so it doesn't always make sense. For
example, three of the solutions reported above give first and
*t* second, but one has the order reversed.

Solving the two equations simultaneously gives us the two solutions we
had before, but it also gives us two solutions we didn't have
before. It often happens that Maple gives you several solutions and
you must decide which (if any) are the ones you want. In the present
case, this is easy because the two unwanted solutions have negative
values of *t* and are not physically important. In fact the two
unwanted solutions come from the first value of `t_hit`, which
we ignored in the the sequential solution.

When the `solve` command is given a set of equations and a set
of variables to solve for, the result of the solve command is one or
more solution sets. To pick out one of the sets, we can use the trick
we used previously of labeling the `solve` command and using
subscripts as shown below.

> sol1 := solve({x(t) = 30, y(t) = 5},{t,theta});

> sol1[2];

Note that if you got the solutions in a different order than shown here, you may need to use an index other than 2 to extract the solution you want.

If you need one of the values from a solution set, the easiest way to
extract it is with the following trick, using the `subs`
command. The trick is to substitute into an expression consisting
simply of the variable whose value is desired. For example, we can get
the value of from the second solution set with the following command.

> subs(sol1[2],theta);

Solving the two equations simultaneously takes less typing, but we did lose some information and flexibility. When we solved the equations seqentially we were able to identify and reject the non-physical solution and, more importantly, plot the distance as a function of . The plot provides a lot of information, including the following points.

- For
*x*larger than about 36.6, there are no solutions. That is, 36.6m is the maximum range. - The maximum range occurs at a value of about 0.84.
- For
*x*smaller than about 17.75 there is only one solution. - For
*x*between these two values, there are two solutions.

Tue Sep 10 11:45:07 EDT 1996