1.

${\displaystyle j={\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle a={\frac {3\pi v_{0}}{2\mathrm {T} }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle v={\frac {v_{0}}{2}}\left(1-\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle x={\frac {v_{0}}{2}}\left(t-{\frac {\mathrm {T} }{3\pi }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle a_{1}=0}$

${\displaystyle v_{1}=v_{0}}$

${\displaystyle x_{1}={\frac {v_{0}\mathrm {T} }{6}}}$

2.

${\displaystyle j=0}$

${\displaystyle a=0}$

${\displaystyle v=v_{0}}$

${\displaystyle x=v_{0}\left(t-{\frac {\mathrm {T} }{6}}\right)}$

${\displaystyle a_{2}=0}$

${\displaystyle v_{2}=v_{0}}$

${\displaystyle x_{2}={\frac {v_{0}\mathrm {T} }{2}}}$

3.

${\displaystyle j=-{\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle a=-{\frac {3\pi v_{0}}{2\mathrm {T} }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle v={\frac {v_{0}}{2}}\left(1+\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle x={\frac {v_{0}}{2}}\left(t+{\frac {\mathrm {T} }{3\pi }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)+{\frac {\mathrm {T} }{3}}\right)}$

${\displaystyle a_{3}=0}$

${\displaystyle v_{1}=v_{0}}$

${\displaystyle x_{\mathrm {f} }={\frac {2v_{0}\mathrm {T} }{3}}}$

Max.

${\displaystyle v_{\max }=v_{0}={\frac {3x_{\mathrm {f} }}{2\mathrm {T} }}}$

${\displaystyle a_{\max }={\frac {3\pi v_{0}}{2\mathrm {T} }}={\frac {9\pi x_{\mathrm {f} }}{4\mathrm {T} ^{2}}}}$

${\displaystyle j_{\max }={\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}={\frac {27\pi ^{2}x_{\mathrm {f} }}{4\mathrm {T} ^{3}}}}$

This media was created with Scilab, a free open-source software. Here is a listing of the Scilab source used to create this file.

function [t1, t2, t3] = indice(t)
    t1 = (t < 1/3);
    t2 = (t >= 1/3) & (t < 2/3);
    t3 = (t >= 2/3);
endfunction

function [j]=jerk(t)
    [t1, t2, t3] = indice(t);
    j = zeros(t);
    j(t1) = (9*%pi^2/2)*cos(3*%pi*t(t1));
    j(t2) = 0;
    j(t3) = -(9*%pi^2/2)*cos(3*%pi*t(t3));
endfunction

function [a]=acceleration(t)
    [t1, t2, t3] = indice(t);
    a = zeros(t);
    a(t1) = (3*%pi/2)*sin(3*%pi*t(t1));
    a(t2) = 0;
    a(t3) = -(3*%pi/2)*sin(3*%pi*t(t3));
    endfunction

function [v] = vitesse(t)
    [t1, t2, t3] = indice(t);
    v = zeros(t);
    v(t1) = (1/2)*(1 - cos(3*%pi*t(t1)));
    v(t2) = 1;
    v(t3) = (1/2)*(1 + cos(3*%pi*t(t3)));
endfunction

function [x] = position(t)
    [t1, t2, t3] = indice(t);
    x = zeros(t);
    x(t1) = (1/2)*(t(t1) - (1/3/%pi)*sin(3*%pi*t(t1)));
    x(t2) = t(t2) - 1/6;
    x(t3) = (1/2)*(t(t3) + (1/3/%pi)*sin(3*%pi*t(t3))) + 1/6;
endfunction

t = 0:0.005:1;

j = jerk(t);
a = acceleration(t);
v = vitesse(t);
x = position(t);

scf(0);
clf;

subplot(4, 1, 1)
plot(t, x)
xtitle("position", "t/T", "$x/(v_0 \mathrm{T})$")

subplot(4, 1, 2)
plot(t, v)
xtitle("vitesse", "t/T", "$v/(v_0)$")

subplot(4, 1, 3)
plot(t, a)
xtitle("accélération", "t/T", "$a\times \mathrm{T}/v_0$")

subplot(4, 1, 4)
xtitle("à-coup", "t/T", "$j \times \mathrm{T}^2/v_0$")
plot(t, j)

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