Mfcttrf

I am trying to get ahead of my upcoming semester by visualizing a differential equation of a general ODE solution for falling objects with the given solution of

$$mfracdvdt = mg - gamma v
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaala
% aabaGaamizaiaadAhaaeaacaWGKbGaamiDaaaacqGH9aqpcaWGTbGa
% am4zaiabgkHiTiabeo7aNjaadAhaaaa!4131!
$$

where the solution is

$$v = (fracmggamma ) + [vo - (fracmggamma )]e^ - fracgamma tm
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2
% da9iaacIcadaWcaaqaaiaad2gacaWGNbaabaGaeq4SdCgaaiaacMca
% cqGHRaWkcaGGBbGaamODaiaad+gacqGHsislcaGGOaWaaSaaaeaaca
% WGTbGaam4zaaqaaiabeo7aNbaacaGGPaGaaiyxaiaadwgadaahaaWc
% beqaaiabgkHiTmaalaaabaGaeq4SdCMaamiDaaqaaiaad2gaaaaaaa
% aa!4CF7!
$$

Where m=mass, g=gravity constant, γ=drag coefficient, v0 is the initial condition, t=time (and is the independent variable)

I would like to create a manipulate and see how different figures for mass, drag, time, vo (maybe?) change the graph or see how the graphs geometrically converge/diverage toward equilibrium solution which is mg/γ but I am not having much success with the manipulate command.

I would appreciate any assistance; I am still getting used to Mathematica.

something to get you started. Set the initial height at 100 above the ground. Plot shows how the object height changes

Manipulate[
 data = Table[i, 
 sol /. m -> theMass, drag -> theDrag, v0 -> theV0, t -> i, i,0, time, 0.01];
 ListLinePlot[data, AxesOrigin -> 0, 100, 
 PlotRange -> 0, 10, 0, 200, AxesLabel -> "Time", "y(t)", 
 BaseStyle -> 14, PlotStyle -> Red, GridLines -> Automatic, 
 GridLinesStyle -> LightGray]
 ,
 theMass, 10, "Mass", 0.01, 10, 0.01, Appearance -> "Labeled",
 theDrag, 1, "Drag", 0.01, 10, 0.01, Appearance -> "Labeled",
 theV0, 20, "Initial velocity", 0, 40, 0.01, 
 Appearance -> "Labeled",
 time, 0.001, "time", 0.001, 10, 0.001, Appearance -> "Labeled",

 TrackedSymbols :> theMass, theDrag, theV0, time,

 Initialization :> 
 g = -9.81;
 ode = m y''[t] == m g - drag y'[t];
 sol = y[t] /. First@DSolve[ode, y[0] == 100, y'[0] == v0, y[t], t]
 
 ]

There is a lot going on in this, especially for a new user, but computers and software seem to have convinced people that they need to build and see things like this and more.

Manipulate[
 sol=v/.DSolve[m v'[t]==m g-gamma v[t],v[0]==v0,v,t][[1]];
 Plot[sol[t],t,0,6],
 m,1,0,2,g,32,16,64,gamma,0,0,1,v0,0,-12,12]

Study the documentation for every part of this and see if you can understand what was the thinking that put this together. You can search for Manipulate and DSolve in the help system and study the examples to see if you can learn how this works. Sometimes clicking on the orange "Details" can provide you with additional information about using a function. You can even search for /. and [[ and == and = to try to understand how each part of this works.