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# In the equation of motion, $S = ut + 1/2a{t^2}$, $S$ stands forA) Displacement in $t$ secondsB) Maximum height reachedC) Displacement in the ${t^{th}}$ secondD) None of these

Last updated date: 13th Jun 2024
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Hint:There are three equations of motions that define motion of an object in one dimension. They are used to derive components of displacement, velocity time and acceleration. The given equation is the second law of motion establishing the relation between displacement, initial velocity, time taken and acceleration of a body.

Formulae used:
Second law of motion: $S = ut + 1/2a{t^2}$.
Where $S$ is the displacement of the body and is expressed in meter $(m)$, $u$ is the initial velocity of the body and is expressed in meter per second $(m/{s^{ - 1}})$, $t$ is the time taken for displacement of the body and is expressed in second $(s)$ and $a$ acceleration of the body and is expressed in meter per second square $(m/{s^{ - 2}})$.
Velocity: $v = \dfrac{{ds}}{{dt}}$
Where $s$ is the displacement of the body and is expressed in meter $(m)$ , $t$ is the time taken for displacement of the body and is expressed in second $(s)$ and $v$ is the final velocity of the body and is expressed in meter per second $(m/{s^{ - 1}})$.
First equation of motion: $v = u + at$
Where $u$ is the initial velocity of the body and is expressed in meter per second $(m/{s^{ - 1}})$ , $t$ is the time taken for displacement of the body and is expressed in second $(s)$ and $v$ is the final velocity of the body and is expressed in meter per second $(m/{s^{ - 1}})$.

Step by step solution:
In Newtonian Mechanics, the main equations that govern and define objects in motion are the three equations of motion. They describe the position, velocity and acceleration of a body, at a given point of time, in a particular frame of reference.
The given equation in the question $S = ut + 1/2a{t^2}$ is known as the “second equation of motion”. It is derived using three methods, namely graphical, algebraic and calculus method. Here we are going to see the calculus method.
We know that velocity is the rate of change of displacement. Mathematically we can write it as $v = \dfrac{{ds}}{{dt}}$.
Upon rearrangement we get,
$v = \dfrac{{ds}}{{dt}} \\ \Rightarrow ds = vdt \\$
But $v = u + at$ according to the first equation of motion.
Substituting the same in the rearranged velocity equation we get,
$ds = (u + at)dt \\ \Rightarrow ds = (udt + atdt) \\$
Considering the body under action of velocity starts from rest, a place of zero displacement and changes its velocity till $t\sec$ we will get the following boundary conditions:
$s:0 \to s \\ t:0 \to t \\$
Applying these limits and integrating the equation $ds = (udt + atdt)$ we get,
$_0{\smallint ^s}ds{ = _0}{\smallint ^t}udt{ + _0}{\smallint ^t}atdt \\ \Rightarrow S = ut + \dfrac{1}{2}a{t^2} \\$
Therefore, we get the second equation of motion that defines displacement, displacement and acceleration of a body with respect to time.

In conclusion, the correct option is A.

Note: The boundary conditions are given for a range of time. Therefore the equation in question can be used to determine the velocity, acceleration or displacement of a body at any instant of time. It is neither for a particular instant like option C or for the maximum height reached like option B.

Additional information:All the equations of Newtonian Mechanics are based on Newton’s Laws of Motion. All equations are a derivation of them and all theories are governed by them. Therefore, Newtonian Mechanics only considers the larger objects on which the gravitational field works.