Answer to Question #152947 in Calculus for Aysha Siddika

Question

The equation is:
v(t)=A(1- e^(-t/t_maxspeed ) )
The tasks are to:
● Use thinking methods to analyse the given engineering problem, e.g. break the problem down into a series of manageable elements, and produce a specification

Answer 1

$v(t)=A(1- e^{-t/t_{maxspeed} } )\\$

If $\pmb{t\rightarrow \infty, v(t)\rightarrow A}$

As, $\lim_{t \rightarrow \infty} v(t)=\lim_{t \rightarrow \infty} A(1-e^{-t/t_{maxspeed}})$

$=A-\lim_{t \rightarrow \infty} e^{-t/t_{maxspeed}}=A-0=A$

For $t=0, v(t)=A-A=0$

$\therefore$ It starts with zero velocity.

Now, acceleration $a(t)=\frac{dv(t)}{dt}=\frac{A}{t_{maxspeed}}e^{-t/t_{maxspeed} }$

If $\pmb{t\rightarrow \infty, a(t)\rightarrow 0}$

As, $\lim_{t\rightarrow \infty}a(t)=\lim_{t\rightarrow \infty}\frac{A}{t_{maxspeed}}e^{-t/t_{maxspeed} }$

$=\frac{A}{t_{maxspeed}}\times0=0$

Now distance travelled at a time $t,$

$d(t)=\int_{0}^{t}v(t)dt=\int_{0}^{t}A(1- e^{-t/t_{maxspeed} } )dt$

$=At+At_{maxspeed}[e^{-t/t_{maxspeed}}-1]$