# Question Video: Finding the Displacement of a Particle Based on Time and Position Relative to a Fixed Point Mathematics

A particle started moving in a straight line. After 𝑡 seconds, its position relative to a fixed point is given by 𝑟 = (𝑡² − 4𝑡 + 7) m, 𝑡 ≥ 0 . Find the displacement of the particle during the first five seconds.

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### Video Transcript

A particle started moving in a straight line. After 𝑡 seconds, its position relative to a fixed point is given by 𝑟 equals 𝑡 squared minus 40 plus seven metres for 𝑡 is greater than or equal to zero. Find the displacement of the particle during the first five seconds.

In this question, we’ve been given a function that describes the position of the particle relative to a fixed point. And we’re being asked to find its displacement. No, it’s absolutely not enough just to substitute 𝑡 equals five into our position function. We recall that if we have a position function of a particle moving along a line given as 𝑟 of t, the displacement from 𝑡 equals 𝑡 one to 𝑡 equals 𝑡 two is the difference between 𝑟 of 𝑡 two minus 𝑟 of 𝑡 one.

This is really important as displacement is the change in the position of the particle. We want to work out the displacement of the particle during the first five seconds. So we’ll let 𝑡 one be equal to zero and 𝑡 two be equal to five. Then the displacement is 𝑟 of five minus 𝑟 of zero. 𝑟 of five is five squared minus four times five plus seven. We simply substitute 𝑡 equals five into our position function. We repeat this process for 𝑟 of zero, this time substituting 𝑡 equals zero in. And we get zero squared minus four times zero plus seven.

That gives us 25 minus 20 plus seven minus seven. Now, seven minus seven is zero. So we’re left with 25 minus 20 which is five. Since our function describes the position relative to a fixed point in metres, we can say that the displacement of our particle during the first five seconds is five metres.