Question Video: Determining When Acceleration Is Zero Based on Time and Velocity | Nagwa Question Video: Determining When Acceleration Is Zero Based on Time and Velocity | Nagwa

Question Video: Determining When Acceleration Is Zero Based on Time and Velocity Mathematics • Third Year of Secondary School

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A particle moves along the 𝑥-axis such that at time 𝑡 seconds its velocity is given by 𝑣 = (𝑡² − 12𝑡 + 3) m/s, 𝑡 ≥ 0. After how many seconds is its acceleration equal to 0?

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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.

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