2) Mass on incline :

Clear["Global`*"]

Part a)

eqs={m1 g Sin[θ] - t1 - u normal == m1 a1,
normal == m1 g Cos[θ],
m2 g - t2 == m2 a2,
r t1 - r t2 == (1/2)m3 (r^2) (a3/r),
a1==-a2,
a3==a1}

Solve[eqs,{t1,t2,normal,a1,a2,a3}] //Simplify

Part b)

eqs2=eqs /.{a1->0,a2->0,a3->0}

eqs2 //TableForm

sol= Solve[eqs2,{t1,t2,normal, u}][[1]] //Simplify

sol //FullSimplify

u /.sol //Together //TrigExpand

((m1 Sin[θ]- m2)/(m1 Cos[θ])) //FullSimplify

3) Falling: linear resistance

Clear["Global`*"]

a) Vertical falling:

eq1= m v'[t] ==m g - b v[t] - c v[t]^2 /.{c->0}

bc=v[0]==v0

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

solT=Solve[eq2, v[t]][[1]]

vTer=v[t] /.solT

vSol=Solve[vT == vTer,b][[1]]

b) Vertical falling:

eqs=Join[{eq1,bc}]

dsol1=DSolve[ eqs ,v[t],t][[1]] //ExpandAll //Simplify

Limit[ v[t] /.dsol1,t-> Infinity,GenerateConditions->False]

dsol2=dsol1 /.vSol //Simplify

v[t] /.dsol2 /.t->0

Limit[ v[t] /.dsol2,t-> Infinity,GenerateConditions->False]

(* Set v0 = to a ratio "r" of vTerminal: *)
values={g->vT,v0-> r vT,vT->1};

velocity[t_,r_]=v[t] /.dsol2 //.values

Plot[{
velocity[t,0],
velocity[t,0.5],
velocity[t,2]},
{t,0,5},
PlotStyle->{Red,Green, Blue},
PlotLegends->{"r=0","r=1/2","r=2"},
PlotRange->All
]

a) Vertical falling: quadratic resistance

eq4= m v'[t] ==m g - b v[t] - c v[t]^2 /.{b->0}

eq5= eq4 /.{v'[t]->0}

solTc=Solve[eq5, v[t]][[1]]

4) Line Integral:

Clear["Global`*"]

Part a

Integrate[2 x,{x,1,0}]+Integrate[3y,{y,0,1}]

Part b

Integrate[2 x,{x,1,0}]+Integrate[3y,{y,0,1}]

rules={fx->2 x - y, fy->x + 3y,x->(((1-t))^2), y -> (t^3),dx->(-2)(1-t)dt, dy-> 3 (t^2)dt}

integrand = fx dx + fy dy //.rules

integrand2=((integrand )/dt)//Expand

Integrate[ integrand2,{t,0,1}]

-4+12/2-9/3+1/5+9/6

5) Planet Hollywood

Clear["Global`*"]

values={m->7.35 (10^22), r->1.74 (10^6),bigG->6.67 (10^(-11))}

Part a)

eq1= m1 g==((bigG m m1 )/(r^2))

solG= Solve[eq1,g][[1]]

g /.solG /.values

(g/9.8) /.solG /.values

Part b)

u = ((- bigG m m1)/r);
u /.values /.{m1->1}

Part c)

eq3= u + ke ==0 /.{ke->(1/2)m1 (v^2)}

solV=Solve[eq3,v][[2]]

vEscape = v /.solV /.values

UnitConvert[Quantity[vEscape,(("Meters")/("Seconds"))],(("Miles")/("Seconds"))]

UnitConvert[Quantity[vEscape,(("Meters")/("Seconds"))],(("Kilometers")/("Seconds"))]

Part d)

eq4= ((bigG m m1)/(r^2)) == m1 ((v^2)/r)

eq5 = v == ((2 π r)/period)

UnitConvert[Quantity[24, "Hours"],"Seconds"]

ruleP= {period ->86400}

solR=Solve[{eq4,eq5},{r,v}][[2]]

r0=r /.solR /.values /.ruleP

Part EXTRA) Use mass of earth

r1=r /.solR /.{m->6 (10^24)} /.values /.ruleP

UnitConvert[Quantity[r1, "Meters"],"Miles"]