A particle moves in such a way that its position vector at any time t is `vec(r)=that(i)+1/2 t^(2)hat(j)+that(k)`.
Find as a function of time:
(i) The velocity `((dvec(r))/(dt))` (ii) The speed `(|(dvec(r))/(dt)|)` (iii) The acceleration `((dvec(v))/(dt))`
(iv) The magnitude of the acceleration
(v) The magnitude of the component of acceleration along velocity (called tangential acceleration)
(v) The magnitude of the component of acceleration perpendicular to velocity (called normal acceleration).

Answer

(i) `hat(i)+that(j)+hat(k)`, (ii) `sqrt(t^(2)+2)`, (iii) `hat(j)`, (iv) `t//sqrt(t^(2)+2)`, (vi) `sqrt(2)//sqrt(t^(2)+2)`

Explanation

`vec(r)=that(i)+t^(2)/2hat(j)+that(k)`
(i) `vec(v)=(dvec(r))/(dt)=hat(i)+that(j)+hat(k)" "` (iii) speed `|vec(v)|=sqrt(t^(2)+2)`
(iii) `vec(a)=(dvec(v))/(dt)=hat(j)" "` (iv) `|vec(a)|=1`
(v) `vec(a)_(T)=(vec(a).hat(v))hat(v)=([hat(j)((hat(i)+that(j)+hat(k)))/(sqrt(t^(2))+2)])((hat(i)+that(j)+hat(k)))/sqrt(t^(2)+2)`
`vec(a)_(T)=(t/sqrt(t^(2)+2))hat(v)=(t(hat(i)+that(j)+hat(k)))/((t^(2)+2)), |vec(a)_(T)|=t/sqrt(t^(2)+2)`
As `a_(N)^(2)+a_(T)^(2)=a^(2)`
so `a_(N)=sqrt(a^(2)-a_(T)^(2))=sqrt(2)/sqrt(t^(2)+2)`