A particle moves along a straight line such that its displacement at any time t is given by $\mathrm{s}=\left({\mathrm{t}}^3-6{\mathrm{t}}^2+3\mathrm{t}+4\right)$ metres. The velocity when the acceleration is zero is

The problem asks to find the velocity of a particle when the acceleration is zero. The displacement of the particle is given as:

s = t³ − 6t² + 3t + 4 meters

To find the velocity when the acceleration is zero, follow these steps:

Step 1: Write Down the Displacement Function

The displacement function of the particle moves along a straight line and is given by:

s = t³ − 6t² + 3t + 4

Step 2: Determine the Velocity Function

The velocity of the particle is the first derivative of the displacement function with respect to time. So, we differentiate the displacement function:

v = ds/dt = d/dt (t³ − 6t² + 3t + 4)

After differentiating, the velocity function becomes:

v = 3t² − 12t + 3

Step 3: Determine the Acceleration Function

The acceleration of the particle is the derivative of the velocity function with respect to time. Therefore, differentiate the velocity function:

a = dv/dt = d/dt (3t² − 12t + 3)

After differentiating, the acceleration function becomes:

a = 6t − 12

Step 4: Set the Acceleration to Zero

To find when the acceleration is zero, set the acceleration function equal to zero:

6t − 12 = 0

Solve for t:

6t = 12

t = 2 seconds

Step 5: Find the Velocity at t = 2 Seconds

Now that we know the particle moves with zero acceleration at t = 2 seconds, we can substitute this value of t into the velocity function to find the velocity at that time:

v = 3t² − 12t + 3

Substitute t = 2 into the velocity equation:

v = 3(2²) − 12(2) + 3

Simplifying the expression:

v = 3(4) − 24 + 3

v = 12 − 24 + 3

v = −9 m/s

Final Answer

The velocity of the particle when the acceleration is zero is −9 m/s.