Answer to Question #267082 in Differential Equations for mary

Question

1. An object has a mass of 2kg is dropped from the top of a building 30 meters tall. the initial velocity is zero. As it falls, the object encounters air resistance that is equal to 1/3 v. what is the velocity and attitude of the object after 1.5 seconds? Ans. 13.02m/s; 19.83m
2. A tank initially contains 50 gal of fresh water. At t=0, brine solution containing 2 lb of salt per gallon is poured into the tank at the rate of 12 gal/min, while the well stirred mixture leaves the tank at the rate of 8 gal/min. what is the amount of salt at the end of 5 minutes? How long will it take to obtain an amount of 50 lb? ans. 88098 lb; 2.47 min

Answer 1

1. The second Newton's Law

$$mv'=mg-\dfrac{1}{3}v$$

Given $m=2\ kg, g=9.81\ m/s^2$

$$v'+\dfrac{1}{6}v=9.81$$

$$v'=-\dfrac{1}{6}(v-58.86)$$

$$\dfrac{dv}{v-58.86}=-\dfrac{1}{6}dt$$

integrate

$$v(t)=58.86+c_1e^{-t/6}$$

$$v(0)=0=>c_1=-58.86$$

$$v(t)=58.86-58.86e^{-t/6}$$

$$v(1.5)=58.86-58.86e^{-1.5/6}$$

$$v(1.5)=13.02\ m/s$$

$$v(t)=-h'(t)$$

$$h(t)=-\int v dt=-\int(58.86-58.86e^{-t/6})dt$$

$$=-58.86t-353.16e^{-t/6}+c_2$$

$$h(0)=30=-353.16+c_2=>c_2=383.16$$

$$h(t)=-58.86t-353.16e^{-t/6}+383.16$$

$$h(1.5)=-58.86(1.5)-353.16e^{-1.5/6}+383.16$$

$$h(1.5)=19.83\ m$$

2. Let $s(t) =$ amount, in lb of salt at time $t.$ Then we have

$\dfrac{ds}{dt}=$(rate of salt into tank) − (rate of salt out of tank)

$$\dfrac{ds}{dt}=2\cdot12-\dfrac{8s}{50+(12-8)t}$$

So we get the differential equation

$$\dfrac{ds}{dt}=24-\dfrac{8s}{50+4t}$$

$$\dfrac{ds}{dt}+\dfrac{4s}{25+2t}=24$$

Integrating factor

$$\mu(t)=e^{\int 4dt/(25+2t)}=(25+2t)^{2}$$

$$\dfrac{d}{dt}((25+2t)^{2}s)=24(25+2t)^{2}$$

$$\int d((25+2t)^{2}s)=24\int(25+2t)^{2}dt$$

$$(25+2t)^{2}s=24(\dfrac{1}{6})(25+2t)^{3}+c_1$$

$$s(t)=4(25+2t)+c_1(25+2t)^{-2}$$

$$s(0)=100+c_1(25)^{-2}=0$$

$$c_1=-62500$$

$$s(t)=4(25+2t)-62500(25+2t)^{-2}$$

$$s(5)=4(25+2(5))-62500(25+2(5))^{-2}$$

$$s(5)=88.98\ lb$$ $$s(t)=4(25+2t)-62500(25+2t)^{-2}=50$$

Let $y(t)=s(t)-50$

$$y(t)=4(25+2t)-62500(25+2t)^{-2}-50$$

Solve graphically the equation $y=0$

$t=2.47\ min$