# SECTION 3.4# Example 1de := m*diff(v(t),t) = m*g-b*v(t);

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dsolve({de,v(0)=v0});

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Int(rhs(%),t);

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value(%);

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sol := x = %+c;

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subs({t=0,x=0},sol);

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solve(%,c);

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subs(c=%,sol);

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# Example 2# Using the result from Example 1,x := m/b*g*t-(v0-m/b*g)/b*m*exp(-b/m*t)+(-m*g+b*v0)*m/b^2;

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x1 := subs({v0=0,m=3,b=3,g=9.81},x);

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fsolve(x1 = 500,t);

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fsolve(9.81*t-9.81 = 500,t); # Ignore exponential term

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plot({x1,500},t=0..60);

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# Example 3#Using x from above,x;

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x1 := subs({v0=0,m=75,b=15,g=9.81},x);

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v1 := diff(x1,t);

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v1_60 := subs(t=60,v1);

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evalf(%);

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x1_60 := subs(t=60,x1);

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evalf(%);

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v1_60 := 49.05;

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x1_60 := 2697.75;

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x2 := subs({t=T,b=105,m=75,v0=v1_60,g=9.81},x);

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fsolve(x2 = 4000-x1_60,T);

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t := 60 + %;

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# Find velocity at impact:v2 := diff(x2,T);

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subs(T=181.56,v2);

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evalf(%);

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# Example 4de := m*diff(v(t),t) = m*g - b*v(t)^r;

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de2 := subs(r=2,de);

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%/m;

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subs(g=v0^2*b/m,%);

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# Solution by separation of variables:LHS := Int(1/(v0^2-v^2),v);

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value(%);

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LHS := %;

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RHS := Int(b/m,t);

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value(%);

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RHS := % + c1;

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sol := LHS = RHS;

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# Solution by dsolve:de2;

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dsolve(de2);

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