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Question: The position \[x\] of a particle varies with time \[\left( t \right)\] as \(x = a{t^2} - b{t^3}\) . ...

The position xx of a particle varies with time (t)\left( t \right) as x=at2−bt3x = a{t^2} - b{t^3} . The acceleration time t of the particle will be equal to zero, where t is equal to
A. 2a3b\dfrac{{2a}}{{3b}}
B. ab\dfrac{a}{b}
C. a3b\dfrac{a}{{3b}}
D. zerozero

Explanation

Solution

We are given a particle which is in motion. We have the equation motion of the particle , so we will find velocity vv and acceleration from the given equation of motion , since v=dxdtv = \dfrac{{dx}}{{dt}} and a=dvdt=d2xdt2a = \dfrac{{dv}}{{dt}} = \dfrac{{{d^2}x}}{{d{t^2}}}, but acceleration is zero hence equating acceleration zero will give us time.

Complete step-by-step answer:
Now from the question,
Velocity: Velocity is defined as the rate of change of its position with respect to a frame of reference with function of time. Velocity is equivalent to a specification of an object's speed and direction of motion.SI unit ms−1m{s^{ - 1}}.
Acceleration: Acceleration can be defined as the rate of change of velocity of an object with respect to time. An object is said to be accelerating if it changes its velocity. It is a vector quantities.SI unit: ms−2m{s^{ - 2}}
Position time graph: In kinematics position time graph, it is the most basic type of graph which describes the motion of the objects. In the position time graph the vertical axis establishes the position of the object and the horizontal axis shows the time elapsed. Hence, the graph tells us where the particle can be found after some amount of time.

Now, we have
Equation of motion x=at2−bt3x = a{t^2} - b{t^3}
Therefore Velocity v=dxdt=2at−3bt2v = \dfrac{{dx}}{{dt}} = 2at - 3b{t^2}
And , Acceleration a=dvdt=d2xdt2=2a−6bta = \dfrac{{dv}}{{dt}} = \dfrac{{{d^2}x}}{{d{t^2}}} = 2a - 6bt
ac=0⇒2a−6bt=0a_c = 0 \Rightarrow 2a - 6bt = 0
t=2a6b=a3bt = \dfrac{2a}{6b} = \dfrac{a}{3b}

Hence, the correct answer is option D - a3b\dfrac{a}{{3b}}

So, the correct answer is “Option D”.

Note: Motion can be served by a position- time graph, which plots the position related to the starting point on the y-axis and the time on the x-axis. The slope of a position- time graph means velocity. The faster the motion is changing the steeper the slope. Average velocity can be estimated from a position- time graph as the change in position divided by the comparable change in time.