Question $2$

A balloon is equipped with temperature and altitude measuring instruments and has radio equipment that can transmit the output readings of these instruments back to the ground. The balloon is initially anchored to the ground with instrument output readings in steady state. The altitude$-$measuring instrument is approximately zero order and the temperature transducer first order with a time constant of $15$ seconds.

Figure Q$2$: First order instrument characteristic

The temperature on the ground, $T_0,$ is $10C$ and the temperature $T_x$ at an altitude of $x$ metres is given by the relation: $T_x=T_{}-0\mathrm{.}01x$

$($a$)$ If the balloon is released at time zero, and thereafter rises upwards at the velocity of

$5$

metres$/$second$,$ draw a table showing the temperature and altitude measurements reported at intervals of $10$ seconds over the first $50$ seconds of travel. Show also in the table the error in each temperature reading.

$[20]$

$($b$)$ What temperature does the balloon report at an altitude of $5000$ metres?

$[5]$