Answer to Question #242926 in Differential Equations for Hassan

Question

A loaded boat weighs 981kg.The force exerted upon the boat in the direct of motion by the oars,is equivalent to a constant force of 15N and the resistance in Newton to the motion is equal to twice the speed of the boat . If th boat starts from rest, determine;

(i)the limiting speed

(ii)speed at the end of 10 sec.

Answer 1

i)

$ma=15-2v$

$m\frac{dv}{dt}=15-2v$

$981\cdot\frac{dv}{dt}=15-2v$

$dt=\frac{981}{15-2v}dv$

$t=\int \frac{981}{15-2v}dv=-\frac{981}{2}ln(15-2v)+c$

$v=0$ at $t=0$

$c=\frac{981}{2}ln15$

$t=-\frac{981}{2}ln(15-2v)+\frac{981}{2}ln15$

$e^{2t/981}=\frac{15}{15-2v}$

$v(t)=7.5-\frac{7.5}{e^{2t/981}}$

ii)

$v(10)=7.5-\frac{7.5}{e^{2\cdot 10/981}}=0.15$ m/s