Page:Calculus Made Easy.pdf/85

(5) Find the acceleration of the body mentioned in the preceding example. Is the acceleration the same for all values of $t$?

(6) The angle $\theta$ (in radians) turned through by a revolving wheel is connected with the time $t$ (in seconds) that has elapsed since starting; by the law

$\theta =2.1-3.2t+4.8t^{2}.$.

Find the angular velocity (in radians per second) of that wheel when $1{\tfrac {1}{2}}$ seconds have elapsed. Find also its angular acceleration.

(7) A slider moves so that, during the first part of its motion, its distance s in inches from its starting point is given by the expression

$s=6.8t^{3}-10.8t$; $t$ being in seconds.

Find the expression for the velocity and the acceleration at any time; and hence find the velocity and the acceleration after $3$ seconds.

(8) The motion of a rising balloon is such that its height $h$, in miles, is given at any instant by the expression $h=0.5+{\tfrac {1}{10}}{\sqrt[{3}]{t-125}}$; $t$ being in seconds.

Find an expression for the velocity and the acceleration at any time. Draw curves to show the variation of height, velocity and acceleration during the first ten minutes of the ascent.

(9) A stone is thrown downwards into water and its depth $p$ in metres at any instant $t$ seconds after reaching the surface of the water is given by the expression

$p={\frac {4}{4+t^{2}}}+0.8t-1$.