(A) Given the position equation: $x = 9t^2 - t^3$. Velocity $v$ is the derivative of position with respect to time: $v = \frac{dx}{dt} = \frac{d}{dt}(

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(A) Given the position equation: $x = 9t^2 - t^3$. Velocity $v$ is the derivative of position with respect to time: $v = \frac{dx}{dt} = \frac{d}{dt}(9t^2 - t^3) = 18t - 3t^2$. To find the time at which speed is maximum,we differentiate velocity with respect to time and set it to zero: $\frac{dv}{dt} = 18 - 6t = 0$. Solving for $t$,we get $t = 3 \ s$. Now,substitute $t = 3 \ s$ into the position equation to find the position at that instant: $x = 9(3)^2 - (3)^3 = 9(9) - 27 = 81 - 27 = 54 \ m$.

Class 11 Physics Motion in Straight Line Instantaneous Velocity and Speed and Velocity-time Graph

Medium

The position $x$ of a particle with respect to time $t$ along the $x$-axis is given by $x = 9t^2 - t^3$,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?

AIPMT 2007 All AIPMT

A
$54$
B
$81$
C
$24$
D
$32$

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Class 11

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Chapter

Motion in Straight Line

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Instantaneous Velocity and Speed and Velocity-time Graph

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The position $x$ of a particle with respect to time $t$ along the $x$-axis is given by $x = 9t^2 - t^3$,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?

Similar Questions

$A$ train moves from one station to another in $2$ hours. Its speed-time graph during this motion is shown in the figure. The maximum acceleration during the journey is ............. $km\, h^{-2}$.

$A$ ball is thrown vertically upwards. Which of the following graphs represents the velocity-time graph of the ball during its flight (air resistance is neglected)?

The position of an object moving along the $x$-axis is given by $x = a + b t^{2}$,where $a = 8.5 \; m$,$b = 2.5 \; m s^{-2}$,and $t$ is measured in seconds. What is the velocity at $t = 0 \; s$ and $t = 2.0 \; s$?

The variation of velocity of a particle moving along a straight line with time is illustrated in the following figure. The distance travelled by the particle in $4 \, s$ is: (in $, m$)

Which of the following graphs represents uniform motion?

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