Question

A particle moves along the $x$-axis so that at time $t\ge 0$ its velocity is given by $v(t)=6t^2-48t-54$. Determine all intervals when the acceleration of the particle is positive.

Step 1 ：Given a particle moving along the x-axis with its velocity function as $v(t)=6t^2-48t-54$ for $t\ge 0$.

Step 2 ：We need to find the intervals when the acceleration of the particle is positive. The acceleration is the derivative of the velocity function.

Step 3 ：Taking the derivative of the velocity function, we get the acceleration function as $a(t)=12t-48$.

Step 4 ：We need to find when this acceleration function is positive, i.e., $a(t)>0$.

Step 5 ：Solving the inequality $12t-48>0$, we get $t>4$.

Step 6 ：Thus, the acceleration of the particle is positive for $t>4$.

Step 7 ：Final Answer: The interval when the acceleration is positive is $\boxed{{\displaystyle (4,\mathrm{\infty })}}$.