The position vector of a particle related to time t is given by → r...

Given, the position vector of the particle is

\overset{\to}{r} = 10t\overset{^}{i} + 15{t}^{2}\overset{^}{j} + 7\overset{^}{k}

The velocity of the particle can be calculated as follows:

\begin{cases}\overset{\to}{v} & = & \dfrac{d\overset{\to}{r}}{dt} \\ & = & \dfrac{d}{dt}\left(10t\overset{^}{i} + 15{t}^{2}\overset{^}{j} + 7\overset{^}{k}\right) \\ & = & 10\overset{^}{i} + 30t\overset{^}{j}\end{cases}

And, the acceleration of the particle is given by

\begin{cases}\overset{\to}{a} & = & \dfrac{d\overset{\to}{v}}{dt} \\ & = & \dfrac{d}{dt}\left(10\overset{^}{i} + 30t\overset{^}{j}\right) \\ & = & 30\overset{^}{j}\end{cases}

Hence, the force on the particle can be written as

\begin{cases}\overset{\to}{F} & = & m\overset{\to}{a} \\ & = & 30m\overset{^}{j}\end{cases}