A particle is moving in a straight line such that its position 𝑟 metres relative to the origin at time 𝑡 seconds is given by 𝑟 equals 𝑡 squared plus three 𝑡 plus seven. Find the particle’s average velocity between 𝑡 equals two seconds and 𝑡 equals four seconds.
Notice how in this example we’re 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 𝑡 equals two seconds and 𝑡 equals four seconds. We, therefore, substitute 𝑡 equals two and 𝑡 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 it’s 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.