Question

If S and x is speed and displacement, respectively, in the expression $S = A(1-e^{-Bxt})$. The unit of B is:
A) $m^{-1} s^{-1}$
B) $m^{-2} s$
C) $s^{-2}$
D) $s^{-1}$

Answer 1

Hint: In physics and metrology, units are the standard for measurement of physical quantities that need clear definitions to be useful. Whenever we need to find the unit we use to explain it with the known SI unit of given terms and accordingly satisfies the equation given.

Complete step by step solution:
Given that, $S = A(1 - {e^{ - Bxt}})$
where S is speed , t is time , x is displacement
Also we know that
unit of speed, $s = \dfrac{{{{metre}}}}{{{{second}}}} $
unit of time, $t = second$
unit of displacement , x = metre
from expression $(1 - {e^{ - Bxt}})$, it can be seen that it has no unit
$\Rightarrow$ unit of $Bxt = 1$
$\Rightarrow B \times x \times t = 1 $
$\Rightarrow B \times metre \times second = 1 $
$\therefore {\text{ unit of B = }}\dfrac{1}{{{{metre \times second}}}} $

Additional Information: Dimensions and units are usually confusing, even though solutions to all engineering problems involve units. Dimensions are physical quantities that can be measured, while units are arbitrary names that relate to particular dimensions to make it relative.
All units for the same dimension are related to each other through a conversion factor, Measurements and calculations require a system of units in which quantities are measured and expressed.
If Q is the unit of derived quantity represented by $Q = M^a L^b T^c$, then $M^a L^b T^c$ is called the dimensional formula and the exponents a, b and c are called dimensions.

Note: Comparing, adding or subtracting quantities with the same dimensions but expressed in different units, the standard procedure is to first convert all of them to the same units. For example, to compare 32 meters with 35 yards, use 1 yard = 0.9144 meters to convert from 35 yards to 32.004 meters.
In total, there are seven primary dimensions(sometimes called basic) dimensions are defined as independent or basic measurements from which other dimensions can be derived. The main dimensions are mass, length, time, temperature, electricity, amount of light, volume of material.