Question

The position x of a particle with respect to time t along x-axis is given by \[x=9{{t}^{2}}-{{t}^{3}}\] where x is in metres and t is in seconds. What will be the position of this particle when it achieves maximum speed along the +x direction?
A. 54m
B. 81m
C. 24m
D. 32m

Answer 1

Hint: The position of any object can find out from the equation of position of that object. To find the velocity we have to differentiate the equation of position. From this, we can find the maximum velocity and the time required to achieve the maximum velocity. From this time period, we can calculate the distance of the particle.

Complete Step-by-Step solution:
As per the question the displacement along the x-axis can be written as,
\[x=9{{t}^{2}}-{{t}^{3}}\], where x is the displacement and t are the time.
We can find the velocity of the particle from the equation of displacement. Since the velocity of the particle is the rate of change of displacement of that particle.
\[v=\dfrac{dx}{dt}\]
We can assign the displacement equation into this.
\[v=\dfrac{d(9{{t}^{2}}-{{t}^{3}})}{dt}\]
\[v=18t-3{{t}^{2}}\]…………………(1)
As we know, the maximum and minimum of anything will occur when the first derivative is zero. So here the maximum velocity will occur when the derivative of velocity becomes zero.
\[\dfrac{dv}{dt}=0\]
We can assign equation (1) into this.
\[\dfrac{d(18t-3{{t}^{2}})}{dt}=0\]
\[18-6t=0\]
\[18=6t\]
\[t=3s\]
So, we found that at 3 seconds the particle will achieve the maximum velocity.
Next, we have to find the displacement of the particle when the velocity is maximum. Therefore, we can plug the time as 3 seconds in the equation of displacement of the particle.
\[x=9\times {{3}^{2}}-{{3}^{3}}\]
\[x=9\times 9-27\]
\[x=54m\]
Therefore, the particle will travel 54 metres to achieve the maximum speed along the positive x-axis. So, the correct answer is option A.

Note: Candidates are advised to remember the condition to find the maximum or minimum of any quantities. The first derivative will help to check the maximum and minimum of any function existing or not. Second derivative can help to identify the maximum and minimum points. Here, we just want to find the time required for the maximum velocity. If the displacement contains a large order of values, we have to do the second derivative also. Otherwise, it will be difficult to find the time.
Since it is a one-dimensional motion, displacement can be treated as the distance and the velocity as the speed.