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.