### Video Transcript

A particle is moving in a straight line such that its position ๐ metres relative to the origin at time ๐ก seconds is given by ๐ equals ๐ก squared plus three ๐ก plus seven. Find the particleโs average velocity between ๐ก equals two seconds and ๐ก equals four seconds.

Notice how in this example weโre given a position for the particle with respect to the origin and asked to find its average velocity. Average velocity is defined as total displacement divided by total time. So rather than differentiating this one, we can simply apply this definition. The displacement will be the total change in position of the particle between ๐ก equals two seconds and ๐ก equals four seconds. We, therefore, substitute ๐ก equals two and ๐ก equals four into the formula for the position of the object and find the difference between these to find the total displacement between two and four seconds.

The position at four seconds is four squared plus three times four plus seven which is 35. And at two seconds itโs two squared plus three times two plus seven which is 17. Between two seconds and four seconds then, the displacement of the particle is 18 metres. The time taken is simply the difference between four seconds and two seconds which is two seconds. And so the average velocity of the particle is 18 over two which is nine metres per second.