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 $t_{2}$ )
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 $t_{2}$ )
$(a)$ False: It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.
$(b)$ True: The arbitrary motion of the particle can be represented by this equation.
$(c)$ False: The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.
$(d)$ False: The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.
$(e)$ True: The arbitrary motion of the particle can be represented by this equation.
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A particle starts from origin at $t=0$ with a velocity $5.0 \hat{ i }\; m / s$ and moves in $x-y$ plane under action of a force which produces a constant acceleration of $(3.0 \hat{ i }+2.0 \hat{ j })\; m / s ^{2} .$
$(a)$ What is the $y$ -coordinate of the particle at the instant its $x$ -coordinate is $84 \;m$ ?
$(b)$ What is the speed of the particle at this time?