A particle executes the motion described by x(t)=x₀(1- ${\mathit{e}}^{-\mathit{\gamma }\mathit{t}}$ );t>0,x_o>0.

(a) Where does the particle start and with what velocity?

(b) Find maximum and minimum values of x (t), v(t),a(t). Show that x (t) and a (t) increase with time and v (t) decreases with time.

<p><span data-teams="true">This is a short answer type question as classified in NCERT Exemplar</span></p><p><strong> (a)&nbsp;</strong>x (t)=x₀ (1-<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mi>γ</mi> <mi>t</mi> </mrow> </mrow> </msup> </math> </span>)</p><p>V (t)=dx/dt= x (t)=x_{0<span contenteditable="false"> <math> <mi>γ</mi> </math> </span>} (<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mi>γ</mi> <mi>t</mi> </mrow> </mrow> </msup> </math> </span>)</p><p>A (t)= x (<span contenteditable="false"> <math> <mi>∞</mi> <mo>, </mo> </math> </span>t)=x₀ (<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mi>γ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mi>γ</mi> <mi>t</mi> </mrow> </mrow> </msup> </math> </span>)</p><p><strong> (b) </strong>when t=o x (t)= x (t)=x₀ (1-<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>0</mn> </mrow> </mrow> </msup> </math> </span>)= x (t)=x₀ (1-1)=0</p><p>x (t) is maximum when t=<span contenteditable="false"> <math> <mi>∞</mi> </math> </span></p><p>x (t) is maximum when t=<span contenteditable="false"> <math> <mn>0</mn> </math> </span></p><p>v (t) is maximum when t=<span contenteditable="false"> <math> <mn>0</mn> </math> </span>, v (0)=x_{0<span contenteditable="false"> <math> <mi>γ</mi> </math> </span>}</p><p>v (t) is maximum when t=<span contenteditable="false"> <math> <mi>∞</mi> <mo>, </mo> </math> </span>&nbsp;v (<span contenteditable="false"> <math> <mi>∞</mi> </math> </span>)=0</p><p>a (t) is maximum when t=<span contenteditable="false"> <math> <mi>∞</mi> </math> </span>, a (<span contenteditable="false"> <math> <mi>∞</mi> </math> </span>)=0</p><p>a (t) is maximum when t=<span contenteditable="false"> <math> <mn>0</mn> </math> </span>, a (<span contenteditable="false"> <math> <mn>0</mn> </math> </span>)=-x_{0<span contenteditable="false"> <math> <mi>γ</mi> </math> </span>}²</p>

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