Position ↔ Velocity ↔ Acceleration ↔ Force

Given the position $\vec{r}$, the velocity $\vec{v}$, the acceleration $\vec{a}$, or the force $\vec{F}$ as a function of time for an object, any other of these four quantities can be calculated.

$\large \vec{r}=\int \vec{v}dt = \int \left( \int \vec{a} dt\right) dt= \int \left(\int \frac{\vec{F}}{m} dt\right) dt,$

$\large \vec{v}=\frac{ d\vec{r}}{dt} = \int \vec{a} dt = \int \frac{\vec{F}}{m} dt, $

$\large \vec{a}=\frac{ d^2\vec{r}}{dt^2} = \frac{ d\vec{v}}{dt}=\frac{\vec{F}}{m}, $

$\large \vec{F}=m\frac{ d^2\vec{r}}{dt^2} = m\frac{ d\vec{v}}{dt}=m\vec{a}. $

Here $m$ is the mass of the object.

Knowing the relationships between $\vec{r}$, $\vec{v}$, $\vec{a}$, and $\vec{F}$ can be quite powerful.

• If an object is known to move in a circle, $\vec{F}=m\frac{ d^2\vec{r}}{dt^2}$ can be used to calculate the centripetal force.
• If an object is moving on the surface of a rotating sphere, $\vec{F}=m\frac{ d^2\vec{r}}{dt^2}$ can be used to calculate the Coriolis force.
• If an object follows a parabolic trajectory, $\vec{F}=m\frac{ d^2\vec{r}}{dt^2}$ can be used to show that the total force on this object is constant.
• If an object travels in a straight line, $\vec{F}=m\frac{ d^2\vec{r}}{dt^2}$ can be used to show that the total force on the object is zero.
• If an object is executing some complicated motion like a stone stuck in a tire or the motion of a part in a machine, $\vec{F}=m\frac{ d^2\vec{r}}{dt^2}$ can be used to calculate the total force on the object.
• If the total force on a satellite is known as well as an initial position and an initial velocity, $\int \left(\int \frac{\vec{F}}{m} dt\right) dt$ can be used to calculate the orbit of the satellite.
• If an electron enters a time dependent electric field, $\int \left(\int \frac{\vec{F}}{m} dt\right) dt$ can be used to calculate the path that this electron follows.
• If an accelerometer is built into a ball, $\vec{F}=m\vec{a}$ can be used to calculate the total force on the ball.
• If a mobile phone is dropped from an airplane, it can transmit its velocity using its accelerometer and $ \vec{v}= \int \vec{a}dt$.

Since these equations can be applied in so many situations, it is not possible to remember the form of the force or the trajectory that an object follows for every case. For the questions in this section: know the equations at the top; determine whether to integrate or differentiate; and derive the solution.