Question

Given the position vector of the particle

r(t)=(t+1)i+(t^2−1)j+2t k, find the particle's velocity and acceleration vectors at t=1

Answer 1

With position vector

`\vec r(t)=(t+1)\,\vec\imath+(t^2-1)\,\vec\jmath+2t\,vec k`

the particle then has velocity

`\vec v(t)=(\mathrm d\vec r(t))/(\mathrm dt)=\vec\imath+2t\,\vec\jmath+2\,vec k`

and acceleration

`\vec a(t)=(\mathrm d\vec v(t))/(\mathrm dt)=(\mathrm d^2\vec r(t))/(\mathrm dt^2)=2\,\vec\jmath`

Then
`t=1`, then particle's velocity and acceleration are, respectively,

`\vec v=\vec\imath+2\vec\jmath+2\,\vec k`

and

`\vec a=2\,\vec\jmath`