Refer to the diagram above. The position of a particle in motion is defined by the vector r(t) (we’ll include its dependence on time for clarity) with respect to some coordinate system/frame of reference. We proceed as follows:

  • Displacement
    • Vector: r(t)=x(t)i+y(t)j+z(t)k (1)
    • Scalar magnitude: r=\sqrt{\left(x(t)\right)^{2}+\left(y(t)\right)^{2}+\left(z(t)\right)^{2}} (2)
  • Velocity (first derivative of displacement)
    • Vector: {\frac{d}{dt}}r(t)=\left({\frac{d}{dt}}x(t)\right)i+\left({\frac{d}{dt}}y(t)\right)j+\left({\frac{d}{dt}}z(t)\right)k (3)
    • Scalar magnitude: v=\sqrt{\left({\frac{d}{dt}}x(t)\right)^{2}+\left({\frac{d}{dt}}y(t)\right)^{2}+\left({\frac{d}{dt}}z(t)\right)^{2}} (4)
  • Acceleration (first derivative of velocity, second derivative of displacement)
    • Vector: {\frac{d^{2}}{d{t}^{2}}}r(t)=\left({\frac{d^{2}}{d{t}^{2}}}x(t)\right)i+\left({\frac{d^{2}}{d{t}^{2}}}y(t)\right)j+\left({\frac{d^{2}}{d{t}^{2}}}z(t)\right)k (5)
    • Scalar magnitude: a=\sqrt{\left({\frac{d^{2}}{d{t}^{2}}}x(t)\right)^{2}+\left({\frac{d^{2}}{d{t}^{2}}}y(t)\right)^{2}+\left({\frac{d^{2}}{d{t}^{2}}}z(t)\right)^{2}} (6)

Now let us look at an example. The motion is defined by the time dependent quantities x(t)=2\,t,\,y(t)=12\,{t}^{2},\,z=10 . Substituting these into the equations above yields the following:

  • Displacement
    • Vector: r(t)=2\,ti+12\,{t}^{2}j+10\,k
    • Scalar magnitude: r=\sqrt{4\,{t}^{2}+144\,{t}^{4}+100}
  • Velocity (first derivative of displacement)
    • Vector: {\frac{d}{dt}}r(t)=2\,i+24\,tj
    • Scalar magnitude: v=\sqrt{4+576\,{t}^{2}}
  • Acceleration (first derivative of velocity, second derivativ
    • Vector: {\frac{d^{2}}{d{t}^{2}}}r(t)=24\,j
    • Scalar magnitude: a=\sqrt{576} = 24