What can be the angle between velocity and acceleration for the motion in two or three dimension ?

A man moves in an open field such that after moving $10 \,m$ on a straight line, he makes a sharp turn of $60^{\circ}$ to his left. The total displacement just at the start of $8^{\text {th }}$ turn is equal to ........$m$

For any arbitrary motion in space, which of the following relations are true

$(a)$ $\left. v _{\text {average }}=(1 / 2) \text { (v }\left(t_{1}\right)+ v \left(t_{2}\right)\right)$

$(b)$ $v _{\text {average }}=\left[ r \left(t_{2}\right)- r \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

$(c)$ $v (t)= v (0)+ a t$

$(d)$ $r (t)= r (0)+ v (0) t+(1 / 2)$ a $t^{2}$

$(e)$ $a _{\text {merage }}=\left[ v \left(t_{2}\right)- v \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

(The 'average' stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

The position vector of a particle changes with time according to the relation $\vec r\left( t \right) = 15{t^2}\hat i + \left( {4 - 20{t^2}} \right)\hat j$. What is the magnitude of the acceleration at $t = 1$ ?

Two particles are projected simultaneously in the same vertical plane, from the same point on ground, but with same speeds but at different angles $( < 90^o )$ to the horizontal. The path followed by one, as seen by the other, is

If vectors $\overrightarrow {A} = cos\omega t\hat i + sin\omega t\hat j$ and $\overrightarrow {B} = cos\frac{{\omega t}}{2}\hat i + sin\frac{{\omega t}}{2}\hat j$ are functions of time, then the value of $t$ at which they are orthogonal to each other is