Plotting Position, Velocity & Acceleration as a function of Time

Topic: Position, Velocity & Acceleration Plot as a Function of Time

Subject: Numerical Methods & Kinematics

Tool: Scilab

by: Gani Comia, Nov. 2024

In the subject of kinematics, the position, velocity, and acceleration as a function of time can be plotted using the forward difference of the numerical method of differentiation. For a given position as a function of time, t, as represented by s = f(t), velocity and acceleration can be solved as s' = f'(t) and s'' = f''(t), respectively.

The Scilab "diff()" function facilitates the numerical differentiation of a given function.

Below is an example of a Scilab script to make a plot of position, velocity, and acceleration as a function of time.

Scilab Script.

// Copyright (C) 2024 - Gani Comia
// Date of creation: 21 Nov. 2024
// Position, Velocity & Acceleration as a function of time
clear;clc;
// Given: Position as a function of time, t.
// s(t)=3t^4 - 35t^3 + 120t^2 - 10

function s=f(t)
    s = 3*t.^4 - 35*t.^3 + 120*t.^2 - 10
endfunction

dt = 0.01                   // time step, dt
t = 0:dt:5                  // time range, t
s = f(t)                    // position, s
v = diff(s)/dt              // velocity, ds/dt (s')
a = diff(s,2)/dt.^2         // acceleration, d2s/dt2 (s'')

clf;
f = gcf()
f.figure_size = [600,700]

// position vs time plot
subplot(3,1,1)
plot(t,s,"-r","linewidth",2.5)
title_1=["Position ,", "$s=3\;t^4-35\;t^3+120\;t^2-10$"]
title(title_1, "fontsize",3)
ylabel("position, s")
xgrid()

// velocity vs time plot
subplot(3,1,2)
plot(t(1:$-1),v,"-b","linewidth",2.5)
title_2=["Velocity ,", "${ds}\;/\;{dt}$"]
title(title_2,"fontsize",3)
ylabel("velocity, v")
xgrid()

// acceleration vs time plot
subplot(3,1,3)
plot(t(1:$-2),a,"-g","linewidth",2.5)
title_3=["Acceleration ,", "${d^2s}\;/\;{dt^2}$"]
title(title_3,"fontsize",3)
ylabel("acceleration, a")
xlabel("time, t","fontsize",2)
xgrid()

Below is the graph as a result of running the Scilab script.

Fig. 1. Graph of Position, Velocity & Acceleration as a function of Time. 
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Disclaimer: The formulas and calculations
presented are for technical reference only. Users must verify the accuracy and
ensure compliance with applicable engineering standards, codes, and safety
requirements before practical application.