Kinematics of Linearly Accelerating Frames

1. Inertial vs. Non-Inertial Frames: A Formal Distinction

Before we derive any equations, we must be precise about what we mean by "inertial" and "non-inertial." An inertial frame is, fundamentally, a frame where Newton's First Law holds true: an object with no net force acting on it moves with a constant velocity. Any frame accelerating with respect to an inertial frame is a non-inertial frame.

The following text provides a formal definition of these concepts, which is the starting point for our entire discussion.

2. The Vector Setup for Linearly Accelerating Frames

As we saw in the reading, the relationship between position vectors in two different frames is the geometric foundation for the transformation equations. Let's formalize this for an accelerating frame.

Consider two reference frames:

• Frame S: An inertial frame (e.g., the ground). We will denote quantities measured in this frame with unprimed variables (e.g., $\vec{r}$, $\vec{v}$, $\vec{a}$).
• Frame S': A non-inertial frame whose origin is accelerating with respect to S. We will denote quantities measured in this frame with primed variables (e.g., $\overrightarrow{r^{\prime }}$, $\overrightarrow{v^{\prime }}$, $\overrightarrow{a^{\prime }}$).

The geometry is described by three position vectors at any given time $t$:

• $\vec{r}(t)$: The position vector of a particle as measured in the inertial frame S.
• $\vec{R}(t)$: The position vector of the origin of the non-inertial frame S' as measured in the inertial frame S.
• $\overrightarrow{r^{\prime }}(t)$: The position vector of the particle as measured in the non-inertial frame S'.

The relationship between these vectors is a simple vector addition, as shown in the diagram below.

From this geometric relationship, we can state our foundational equation:
$\vec{r}(t)=\vec{R}(t)+\overrightarrow{r^{\prime }}(t)$
This equation simply states that the position of the particle with respect to the inertial origin is the sum of the position of the non-inertial origin and the particle's position with respect to that non-inertial origin.

3. Derivation of the Transformation Equations

With our vector relationship established, the derivation of the kinematic transformation equations is a direct application of calculus. Since you prefer rigorous derivations, let's turn to a resource that presents it cleanly and directly.

Let's carefully re-trace the logic from the reading, ensuring every step is clear.

Position Transformation

This is our starting point, based on the vector diagram:
$\vec{r}(t)=\vec{R}(t)+\overrightarrow{r^{\prime }}(t)$

Velocity Transformation

To find the relationship between velocities, we differentiate the position equation with respect to time. Using the Leibniz notation for clarity:
$\frac{d}{dt}\vec{r}(t)=\frac{d}{dt}\vec{R}(t)+\frac{d}{dt}\overrightarrow{r^{\prime }}(t)$
Each of these time derivatives is a velocity:

• $\vec{v}(t)=\frac{d\vec{r}}{dt}$ is the velocity of the particle in the inertial frame S.
• $\vec{V}(t)=\frac{d\vec{R}}{dt}$ is the velocity of the non-inertial frame S' with respect to S.
• $\overrightarrow{v^{\prime }}(t)=\frac{d\overrightarrow{r^{\prime }}}{dt}$ is the velocity of the particle in the non-inertial frame S'.

Substituting these definitions gives the velocity transformation equation:
$\vec{v}(t)=\vec{V}(t)+\overrightarrow{v^{\prime }}(t)$
This is the familiar law of addition of velocities, which holds true whether the frames are inertial or not.

Acceleration Transformation

To find the core result, we differentiate the velocity equation with respect to time:
$\frac{d}{dt}\vec{v}(t)=\frac{d}{dt}\vec{V}(t)+\frac{d}{dt}\overrightarrow{v^{\prime }}(t)$
Again, each term is an acceleration:

• $\vec{a}(t)=\frac{d\vec{v}}{dt}$ is the acceleration of the particle in the inertial frame S.
• $\vec{A}(t)=\frac{d\vec{V}}{dt}$ is the acceleration of the non-inertial frame S' with respect to S. By definition, for a non-inertial frame, $\vec{A}\ne \vec{0}$.
• $\overrightarrow{a^{\prime }}(t)=\frac{d\overrightarrow{v^{\prime }}}{dt}$ is the acceleration of the particle as measured in the non-inertial frame S'.

Substituting these gives the acceleration transformation equation:
$\vec{a}(t)=\vec{A}(t)+\overrightarrow{a^{\prime }}(t)$
This is the most important result of today's lesson. It shows that the acceleration measured in an inertial frame ($\vec{a}$) is the sum of the acceleration of the non-inertial frame itself ($\vec{A}$) and the acceleration measured within that non-inertial frame ($\overrightarrow{a^{\prime }}$).

4. The Origin of Fictitious Forces

Your goal is to understand the "why" behind physics, so let's briefly look at the profound consequence of this kinematic result for dynamics.

In the inertial frame S, Newton's Second Law is simple:
${\vec{F}}_{\text{net}}=m\vec{a}$
where ${\vec{F}}_{\text{net}}$ represents the sum of all real physical forces (gravity, normal force, tension, etc.).

Now, substitute our acceleration transformation into this law:
${\vec{F}}_{\text{net}}=m(\vec{A}+\overrightarrow{a^{\prime }})$
If we rearrange this equation to solve for the term $m\overrightarrow{a^{\prime }}$, which looks like "mass times acceleration in the non-inertial frame," we get:
$m\overrightarrow{a^{\prime }}={\vec{F}}_{\text{net}}-m\vec{A}$

An observer in the non-inertial frame S' measures the acceleration $\overrightarrow{a^{\prime }}$. To them, it appears that the effective force causing this acceleration is ${\vec{F}}_{\text{net}}-m\vec{A}$. They observe the real forces ${\vec{F}}_{\text{net}}$, but they also perceive an additional term, ${\vec{F}}_{\text{fictitious}}=-m\vec{A}$. This is the fictitious force or inertial force.

It's not a real force caused by an interaction. It is a direct mathematical consequence of observing motion from an accelerating frame of reference. This is why you feel pushed back into your seat when a car accelerates forward: your frame of reference (the car) is accelerating forward ($\vec{A}$), so you experience a fictitious force in the opposite direction ($-m\vec{A}$).

Conclusion

In this lesson, we have formally extended our study of relative motion to include linearly accelerating reference frames. By starting with the simple vector addition of position vectors and applying differentiation, we rigorously derived the kinematic transformation laws.

Key Takeaways:

• An inertial frame is one where Newton's First Law holds. A non-inertial frame accelerates with respect to an inertial frame.
• The fundamental kinematic transformation equations for a frame S' moving with acceleration $\vec{A}$ relative to an inertial frame S are:
  • Position: $\vec{r}=\vec{R}+\overrightarrow{r^{\prime }}$
  • Velocity: $\vec{v}=\vec{V}+\overrightarrow{v^{\prime }}$
  • Acceleration: $\vec{a}=\vec{A}+\overrightarrow{a^{\prime }}$
• The acceleration transformation, when inserted into Newton's Second Law, gives rise to an apparent fictitious force, ${\vec{F}}_{\text{fictitious}}=-m\vec{A}$, which must be accounted for by an observer in the non-inertial frame.

Preview of the Next Lesson:
We have derived the necessary equations. In the next lesson, we will put them into practice. We will apply the derived transformation equations to analyze the trajectory of an object as observed from a non-inertial frame, solving problems that require you to switch perspectives between inertial and non-inertial observers.