Video Transcript
A particle moves along the 𝑥-axis
such that at time 𝑡 seconds its velocity is given by 𝑣 is equal to 𝑡 squared
minus 12𝑡 plus three meters per second, where 𝑡 is greater than or equal to
zero. After how many seconds is its
acceleration equal to zero?
In this question, we are given an
expression for the velocity of a particle at 𝑡 seconds. It is 𝑡 squared minus 12𝑡 plus
three meters per second. To find its acceleration, we recall
the link between velocity and acceleration: 𝑎 is equal to d𝑣 by d𝑡. This means that acceleration is the
change in velocity with respect to time. We need to differentiate our
expression for velocity with respect to time. And we will do this term by
term.
The derivative of 𝑡 squared with
respect to 𝑡 is two 𝑡. Differentiating negative 12𝑡 with
respect to 𝑡, we get negative 12. Finally, when differentiating a
constant, we get zero. So, d𝑣 by d𝑡 is equal to two 𝑡
minus 12. And our expression for the
acceleration of the particle is also equal to two 𝑡 minus 12.
We are asked to calculate after how
many seconds the acceleration is equal to zero. This means we need to set our
expression equal to zero. We can solve for 𝑡 by firstly
adding 12 to both sides of the equation. We then divide both sides of the
equation by two, giving us 𝑡 is equal to six. And we can therefore conclude that
the acceleration of the particle is equal to zero after six seconds.
