Question Video: Finding the Displacement Expression of a Body given Its Position Expression Relative to Time Mathematics

If the position vector of a body at time 𝑡 is given by 𝐫(𝑡) = (−3𝑡² − 5)𝐢 + (−4𝑡 − 6)𝐣, find its displacement 𝐬(𝑡).

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Video Transcript

If the position vector of a body at time 𝑡 is given by 𝐫 of 𝑡 is equal to negative three 𝑡 squared minus five 𝐢 plus negative four 𝑡 minus six 𝐣, find its displacement 𝐬 of 𝑡.

We recall that the displacement of a body is its change in position. It is the distance from the start point or origin. We can calculate the displacement vector 𝐬 of 𝑡 by subtracting 𝐫 of zero, the initial point, from 𝐫 of 𝑡. To calculate 𝐫 of zero, the initial position, we substitute zero into our equation. This gives us negative three multiplied by zero squared minus five 𝐢 plus negative four multiplied by zero minus six 𝐣. Simplifying this gives us negative five 𝐢 minus six 𝐣.

We now need to subtract this vector from 𝐫 of 𝑡. 𝐬 of 𝑡 is therefore equal to negative three 𝑡 squared minus five 𝐢 plus negative four 𝑡 minus six 𝐣 minus negative five 𝐢 minus six 𝐣. Negative five 𝐢 minus negative five 𝐢 is equal to zero. Likewise, negative six 𝐣 minus negative six 𝐣 is also zero. This means that the vector for displacement 𝐬 of 𝑡 is equal to negative three 𝑡 squared 𝐢 plus negative four 𝑡 𝐣.

You might notice at this stage that the only difference between the position vector 𝐫 of 𝑡 and the displacement vector 𝐬 of 𝑡 is that the constants have disappeared. This will always be the case as these constants will represent the starting position of the body, in this case, negative five 𝐢 minus six 𝐣.

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