Problem 1

Clear["Global`*"]

Vertical falling:

eq1= m v'[t] ==m g - b v[t] - c v[t]^2

bc=v[0]==v0

eq2= eq1 /.{v'[t]->0}

eq2b= eq2 /.{c->0}

eq2c=eq2 /.{b->0}

bSol=Solve[eq2b,v[t]][[1]]

cSol=Solve[eq2c,v[t]][[2]]

values={b->β D, c -> γ (D^2),β->1.6 (10^(-4)),γ->0.25};

vb=v[t] /.bSol //.values /.{g->10, m->100 ,D->1}

vc= v[t] /.cSol //.values /.{g->10, m->100 ,D->1}

Needs["Units`"]

Convert[vb Meter/Second,Mile/Hour]

Convert[vc Meter/Second,Mile/Hour]

Problem 2

Horizontal

Clear["Global`*"]

(* To start, we'll zero gravity  *)
eq1= m v'[t] == m g - b v[t] - c v[t]^2 /.{g->0}

bc=v[0]==v0  (* Boundary Conditions *)

dsol1=DSolve[{eq1,bc} ,v[t],t][[1]] //FullSimplify

(* Look at ONLY LINEAR term *)
dsol2=DSolve[{eq1,bc} /.{c->0},v[t],t][[1]]

(* Look at ONLY QUADRATIC term *)
dsol3=DSolve[{eq1,bc} /.{b->0},v[t],t][[1]]

v[t] /.dsol1

v[t] /.dsol2

v[t] /.dsol3

(* Pick some values *)
values={v0->1, m->1};

v[t] /.dsol1 /.values

tmp1=v[t] /.{dsol1,dsol2,dsol3} /.values

Plot[tmp1 /.{b->1, c->1} //Evaluate,{t,0,10},
PlotStyle->{Red,Green,Blue},PlotLegends->LineLegend[{Red,Green,Blue},{"Gen","c=0","b=0"}]]

fun[b_,c_]=v[t] /.{dsol1,dsol2,dsol3} /.values

?Manipulate

Manipulate[
Plot[fun[b,c] //Evaluate,{t,0,10},
PlotStyle->{{Thickness[0.015],Red},Green,Blue},
PlotLabel->{"b=",b, "   c= ",c},
PlotLegends->LineLegend[{Red,Green,Blue},{"Both","Linear","Quadratic"}],
PlotRange->{Automatic,{0,1}}
]
,{{b,1,"Linear"},0.1,2,0.1},{{c,1,"Quadratic"},0.1,2,0.1}]

Vertical
Now add gravity

Clear["Global`*"]

(* To start, we'll zero gravity  *)
eq1= m v'[t] == m g - b v[t] - c v[t]^2

bc=v[0]==v0  (* Boundary Conditions *)

dsol1=DSolve[{eq1,bc} ,v[t],t][[1]] //FullSimplify

(* Look at ONLY LINEAR term *)
dsol2=DSolve[{eq1,bc} /.{c->0},v[t],t][[1]] //FullSimplify

(* Look at ONLY QUADRATIC term *)
dsol3=DSolve[{eq1,bc} /.{b->0},v[t],t][[1]]//FullSimplify

v[t] /.dsol1

v[t] /.dsol2

v[t] /.dsol3

(* Pick some values *)
values={ m->1,g->1};

v[t] /.dsol1 /.values

tmp1=v[t] /.{dsol1,dsol2,dsol3} /.values

Plot[tmp1 /.{v0->0}/.{b->1, c->1} //Evaluate,{t,0,10},
PlotStyle->{Red,Green,Blue},PlotLegends->LineLegend[{Red,Green,Blue},{"Gen","c=0","b=0"}]]

fun[b_,c_,v0_]=v[t] /.{dsol1,dsol2,dsol3} /.values

?Manipulate

Manipulate[
Plot[fun[b,c,v0] //Evaluate,{t,0,10},
PlotStyle->{{Thickness[0.015],Red},Green,Blue},
PlotLabel->{"b=",b, "   c= ",c},
PlotLegends->LineLegend[{Red,Green,Blue},{"Both","Linear","Quadratic"}],
PlotRange->{Automatic,{0,2}}
],{{b,1,"Linear"},0.1,2,0.1},
{{c,1,"Quadratic"},0.1,2,0.1},
{{v0,0},0,2,0.2}
]

Problem 3

Clear["Global`*"]

Part 1 :

eq1= {m  a == m g Sin[θ]   - μ normal, normal == m g Cos[θ] };

eq1

sol= Solve[eq1,{a,normal}][[1]]

sol  /.{θ->π/2}

work = μ normal d /.sol

vSol= Solve[ 2 a d == (v^2),v][[2]]

kinetic = (1/2) m (v^2) /.vSol /.sol //Expand

Part 2 :

eq= energy == m g h - μ normal d  /.{h -> d Sin[θ]} /.sol

eq= eq  /.{energy-> (1/2)m (v^2)}

Solve[eq,v]

Problem 5

Clear["Global`*"]

PlanetData[]

mass=PlanetData["Earth","Mass"]

rTmp=PlanetData["Earth","Radius"]

radius=UnitConvert[rTmp,"SI"]

radius=UnitConvert[radius,"Meter"]

momInertia = (2/5) mass (radius^2)

day=UnitConvert[Quantity[1,"Days"],"Seconds"] //N

omega=2 π /day

angMom= momInertia omega

angAcc= omega/ day

torque= momInertia angAcc

force = torque/radius

UnitConvert[force,"Newton"]