5 Engineering Example 4

5.1 Velocity of a rocket

The upward velocity of a rocket, measured at 3 different times, is shown in the following table

The velocity over the time interval $5\le t\le 12$ is approximated by a quadratic expression as

$v\left(t\right)=a_1{t}^2+a_{2}t+a_3$

Find the values of $a_1,a_2$ and $a_3$ .

Solution

Substituting the values from the table into the quadratic equation for $v\left(t\right)$ gives:

Applying one of the methods from this Workbook gives the solution as

$a_1=0.2905a_2=19.6905a_3=1.0857$ to 4 d.p.

As the original values were all experimental observations then the values of the unknowns are all approximations . The relation $v\left(t\right)=0.2905t^2+19.6905t+1.0857$ can now be used to predict the approximate position of the rocket for any time within the interval $5\le t\le 12$ .

Exercises

Solve the following using Gauss elimination:

1.

$\begin{array}{ccccccc}\hfill 2x_1& \hfill +& \hfill x_2& \hfill -& \hfill x_3& \hfill =& \hfill 0\\ \hfill x_1& \hfill & \hfill & \hfill +& \hfill x_3& \hfill =& \hfill 4\\ \hfill x_1& \hfill +& \hfill x_2& \hfill +& \hfill x_3& \hfill =& \hfill 0\end{array}$

2.

$\begin{array}{ccccccc}\hfill x_1& \hfill -& \hfill x_2& \hfill +& \hfill x_3& \hfill =& \hfill 1\\ \hfill -x_1& \hfill & \hfill & \hfill +& \hfill x_3& \hfill =& \hfill 1\\ \hfill x_1& \hfill +& \hfill x_2& \hfill -& \hfill x_3& \hfill =& \hfill 0\end{array}$

3.

$\begin{array}{ccccccc}\hfill x_1& \hfill +& \hfill x_2& \hfill +& \hfill x_3& \hfill =& \hfill 2\\ \hfill 2x_1& \hfill +& \hfill 3x_2& \hfill +& \hfill 4x_3& \hfill =& \hfill 3\\ \hfill x_1& \hfill -& \hfill 2x_2& \hfill -& \hfill x_3& \hfill =& \hfill 1\end{array}$

4.

$\begin{array}{ccccccc}\hfill x_1& \hfill -& \hfill 2x_2& \hfill -& \hfill 3x_3& \hfill =& \hfill -1\\ \hfill 3x_1& \hfill +& \hfill x_2& \hfill +& \hfill x_3& \hfill =& \hfill 4\\ \hfill 11x_1& \hfill -& \hfill x_2& \hfill -& \hfill 3x_3& \hfill =& \hfill 10\end{array}$

You may need to think carefully about this system.

Answer