1. The problem is to understand the motion of a particle under a constant force, such as gravity, which causes constant acceleration. 2. The fundamental formula used is Newton's second law: $$\vec{F} = m\vec{a}$$ where $\vec{F}$ is the constant force, $m$ is the mass, and $\vec{a}$ is the acceleration. 3. Since the force is constant, acceleration is constant: $$\vec{a} = \frac{\vec{F}}{m}$$. 4. Acceleration is the second derivative of position with respect to time: $$\vec{a} = \frac{d^2 \vec{r}}{dt^2}$$. 5. Integrating acceleration once with respect to time gives velocity: $$\vec{v}(t) = \vec{v}_0 + \vec{a}t$$ where $\vec{v}_0$ is the initial velocity. 6. Integrating velocity gives position: $$\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2$$ where $\vec{r}_0$ is the initial position. 7. For example, under gravity near Earth's surface, $\vec{F} = m\vec{g}$, so $\vec{a} = \vec{g}$ is constant downward acceleration. 8. This explains the familiar equations of motion for free-fall or projectile motion under constant gravity. Final answer: The position as a function of time under a constant force is $$\boxed{\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \frac{\vec{F}}{m} t^2}$$.