Question

A body moves according to the formula$v{\text{ }} = {\text{ }}1{\text{ }} + {\text{ }}{t^2}$ where v is the velocity at time t. The acceleration after 3 sec will be (v in cm/sec)
A. 24$m{s^{ - 2}}$
B. 12$m{s^{ - 2}}$
C. 6$m{s^{ - 2}}$
D. None of the above

Answer 1

The rate of change of an object's velocity with respect to time is called acceleration in mechanics. Accelerations are values that are measured in vectors (in that they have magnitude and direction). The orientation of the net force applied on an item determines the orientation of its acceleration. The metre per second squared is the SI unit for acceleration.

Formula used:
${\mathbf{a}} = \dfrac{{d{\mathbf{v}}}}{{dt}}$
A = acceleration
dv = change in velocity
dt = change in time

Complete step by step answer:
Newton's Second Law describes the magnitude of an object's acceleration as the combined impact of two sources.
The amount of this net resultant force is exactly proportional to the net balance of all external forces acting on that item.
an object's mass, which varies based on the materials used in its construction – magnitude is inversely related to mass.
Now from the given question
The derivative of a function of a real variable in mathematics indicates the sensitivity of the function result (output value) to changes in its argument (input value). Calculus uses derivatives as a fundamental technique. The velocity of a moving item, for example, is the derivative of its position with respect to time: it quantifies how rapidly the object's position changes as time passes.
$v{\text{ }} = {\text{ }}1{\text{ }} + {\text{ }}{t^2}$
Upon differentiating it
$\dfrac{dv}{dt} = 0 + 2t$
$\Rightarrow \dfrac{dv}{dt} = 2t$
Now substituting the value of t in the formula we get
$a = 2(3)$
$\Rightarrow a = 6m{s^{ - 2}}$

So, the correct answer is “Option A”.

Note:
Note that here we can’t use the formula $a = \dfrac{{(v - u)}}{t}$since the required inputs are not given. Meanwhile, instantaneous acceleration is the maximum average acceleration over an infinitesimal time span. Instantaneous acceleration is the derivative of the velocity vector with respect to time in calculus terminology. Because both acceleration and deceleration are changes in velocity, they are regarded the same. Passengers feel each of these accelerations (tangential, radial, and deceleration) until their relative (differential) velocity in relation to the vehicle is neutralised.