Question Video: Finding the Displacement of a Particle Based on Time and the Position Vector of the Particle Mathematics

The position vector of a particle relative to the point 𝑂 is given by the relation 𝐫 = (𝑑² + 4𝑑 βˆ’ 5)𝐒, where 𝐒 is a fixed unit vector and 𝑑 is the time. Find the displacement of the particle after 3 seconds.

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Video Transcript

The position vector of a particle relative to the point 𝑂 is given by the relation 𝐫 is equal to 𝑑 squared plus four 𝑑 minus five 𝐒, where 𝐒 is a fixed unit vector and 𝑑 is the time. Find the displacement of the particle after three seconds.

The displacement of a particle is the change in position. This is measured from the origin or start point. Therefore, the displacement of the particle after three seconds will be equal to 𝐫 of three minus 𝐫 of zero. When 𝑑 is equal to three, 𝐫 is equal to three squared plus four multiplied by three minus five 𝐒. Three squared is equal to nine. Four multiplied by three is 12. So we have nine plus 12 minus five 𝐒. Nine plus 12 is equal to 21, and subtracting five gives us 16𝐒.

When 𝑑 is equal to zero, we have zero squared plus four multiplied by zero minus five 𝐒. Both zero squared and four multiplied by zero are equal to zero. This means we’re left with negative five 𝐒. We need to subtract this from 16𝐒. This is the same as adding five 𝐒 to 16𝐒. The displacement of the particle after three seconds is therefore equal to 21𝐒.

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