Question

The dimensional formula of distance travelled in ${n^ {th}} $ second by a particle moving with uniform acceleration are,
A) $[{M^0}{L^1}{T^{ - 1}}]$
B) $[{M^0}{L^1}{T^0}]$
C) $[{M^0}{L^0}{T^{ - 1}}]$
D) $[{M^1}{L^1}{T^1}]$

Answer 1

Hint: Use the equations of motion to calculate the dimensional formula for the distance travelled. These equations of motion describe the basic concept of the motion of an object such as the velocity, acceleration and the position of an object at various instances of times. Also remember that there are three equations of motion.

Complete solution:
A dimension analysis is conducted to analyse the relationships between the various physical quantities by defining their specific quantities (like length, weight, time and electric load) and units of measures (like miles vs. kilometres, and pounds vs. kilos). Due to the usual 10-base of both of the units, converting units from one unit to another within the metric or SI system is always simpler than others. The unit-factor method, or more simply the factor-label method, is also called unit-factor method.

Physically equivalent sums (also known as "commensurable") (e.g. length, duration, or mass) shall be of the same size and can, even though initially represented in separate units of measures (e.g. yards, and yards), be contrasted similarly with other physical quantities of the same sort. If physical quantities have different measures (for example, length vs. weight), the quantities cannot be compared in terms of equivalent units (also called measurable). For starters, it is pointless to question if a kilogramme reaches an hour.

Any physically important equation (and any inequality) on the left and right sides of a property called dimensional homogeneity would have the same dimensions. Dimensional homogeneity control is a standard application for dimensional analysis which serves as a plausibility check for derived equations and calculations. It also acts as a reference and restriction on deriving equations which without more rigorous derivations, can define a physical structure.
Distance travelled in the ${n^ {th}} $ second by a particle moving with uniform acceleration,
Using equation of motion,
$s = u + \dfrac {a} {2} (2n - 1) $
$u$ is the velocity, therefore, all the dimensions will be in velocity.

Hence, $[{M^0} {L^1} {T^ {0}}] $ is the dimensional formula of distance travelled in ${n^ {th}} $ second by a particle moving with uniform acceleration.

Note: Be careful while using the time for any defined second. In physics, the equations of the motion are defined as the equations which describes the behaviour of a physical system as a function of time in terms of its motion. These three equations of motion govern the motion of an object in 1D, 2D and 3D.