Calculation of true thickness of the bed

 104. The Laplace Transform of unit step function is:

 (a) 1         

(b) $\frac{1}{s}$       

(c) $\frac{1}{s^{2}}$

(d) $\frac{1}{s^{3}}$

 

(Thanks to Pragnath, AU)

 

Solution: -

The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain.  if x(t) is a time-domain function, then its Laplace transform is defined as

$X_{(s)} = L[u(t)] = \int_{0}^{\infty}~u(t)e^{-st}~dt$

 

The unit step function is defined as,

 $u(t) = 0,$  for $t < 0$

              $1,$ for t ≥ 0

Therefore, by the definition of the Laplace transform, we get,

$X_{(s)} = L[u(t)] = \int_{0}^{\infty}~u(t)e^{-st}~dt$

 

$\Rightarrow~L[u(t)]~=~\int_{0}^{\infty} e^{-st}~dt~=~[\frac{e^{-st}}{-s}]_0^\infty$     

$\Rightarrow~L[u(t)]~=~[\frac{e^{\infty}-e^0}{-s}]~=~\frac{1}{s}$

 

Therefore the answer is option   (b)   $\frac{1}{s}$