Given the position of a mass, can we find its velocity and acceleration?

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The velocity is the rate of change of position and the acceleration is the rate of change of velocity, so we find them by differentiating with respect to time. \[\begin{align*} \text{Velocity, }\mathbf{v} &= \dfrac{d\mathbf{r}}{dt} = (6t+2)\mathbf{i}-8t\,\mathbf{j} \\ \text{Acceleration, }\mathbf{a} &= \dfrac{d\mathbf{v}}{dt}=6\mathbf{i}-8\mathbf{j}. \end{align*}\]

Newton’s second law tells us that the force, \(\mathbf{F}=m\,\mathbf{a}\). The acceleration and therefore the force are constant. So we have (in \(\mathrm{N}\)) \[\mathbf{F} = 3 \times (6\mathbf{i}-8\mathbf{j}) = 6(3\mathbf{i}-4\mathbf{j}).\]

So the magnitude of the force is \[6\sqrt{3^2+4^2} = \quantity{30}{N}\] as required.

From the diagram, \(\tan\theta=\frac{8}{6}\) so the direction of the force is \(53.1^\circ\) below the \(\mathbf{i}\)-direction.