A particle moves along the 𝑥-axis such that at time 𝑡 seconds, its velocity is given by 𝑣 equals 𝑡 squared minus 12𝑡 plus three meters, for 𝑡 is greater than or equal to zero. After how many seconds is its acceleration equal to zero?
In this question, we’re given an expression for the velocity of the particle at 𝑡 seconds. It’s 𝑡 squared minus 12𝑡 plus three. To find its acceleration, we recall the link between velocity and acceleration. Acceleration is defined as the rate of change of velocity, so change in velocity with respect to time. In other words, we differentiate an expression for velocity with respect to time to find an expression for acceleration.
Our expression for velocity includes three terms, so we’ll differentiate term by term. The derivative of 𝑡 squared with respect to 𝑡 is two 𝑡. Then we differentiate negative 12𝑡 with respect to 𝑡. And we get negative 12. Finally, when we differentiate a constant, we get zero. So an expression for acceleration at time 𝑡 is two 𝑡 minus 12.
This question’s asking us after how many seconds is the acceleration of the particle equal to zero. So we’re going to set this equal to zero and solve for 𝑡. We add 12 to both sides of this equation. So two 𝑡 equals 12. Finally, we’ll divide both sides of our equation by two. So 𝑡 is 12 divided by two, which is equal to six. And we can, therefore, say that the acceleration of the particle is zero after six seconds.