# Video: Determining the Velocity given the Displacement Expression of a Particle

A particle moves along the 𝑥-axis such that at time 𝑡 seconds, its displacement from the origin is given by 𝑠 = (𝑡³ − 4𝑡²) m, 𝑡 ≥ 0. What is the particle’s average velocity in the first 10 seconds?

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### Video Transcript

A particle moves along the 𝑥-axis such that at time 𝑡 seconds, its displacement from the origin is given by 𝑠 is equal to 𝑡 cubed minus 4𝑡 squared metres, where 𝑡 is greater than or equal to zero. What is the particle’s average velocity in the first 10 seconds?

Our first thought in a question like this might be to differentiate the expression for displacement to get an expression for velocity. However, in this question, we need to calculate the average velocity. This can be calculated by dividing the total displacement by the total time. We need to calculate the average velocity in the first 10 seconds. This means that the total time is 10 seconds.

Initially, when 𝑡 equals zero, the displacement is equal to zero cubed minus four multiplied by zero squared. This is equal to zero metres as the particle starts at the origin. At 𝑡 equals 10, then the displacement is equal to 10 cubed minus four multiplied by 10 squared. 10 cubed is equal to 1000. 10 squared is equal to 100. And multiplying this by four gives us 400. 1000 minus 400 is equal to 600. Therefore, the displacement at time 10 seconds is 600 metres.

We can calculate the total displacement by subtracting these two numbers: 600 metres and zero metres. This gives us 600 metres. The average velocity can, therefore, be calculated by dividing 600 by 10. This is equal to 60. The displacement is measured in metres and the time in seconds. Therefore, the velocity will be measured in metres per second.

We can, therefore, conclude that the average velocity of the particle in the first 10 seconds is 60 metres per second.