3: Relative motion analysis (translating axes)

Relative Motion Analysis (Translating Axes)

In particle kinematics, relative motion analysis using translating axes simplifies problems where multiple objects move independently. This method isolates the motion of one particle relative to another by attaching a moving coordinate system (frame) to a reference particle. Crucially, the frame translates but does not rotate, meaning its axes remain parallel to the fixed global axes.

Key Concepts

1. Position Vectors:

   • Let $\vec{r_A}$ and $\vec{r_B}$ be the positions of particles $A$ and $B$ relative to a fixed frame $O$.
   • The position of $A$ relative to $B$ is $\vec{r_{A/B}} = \vec{r_A} - \vec{r_B}$.

2. Velocity Relationship:

   • The velocity of $A$ is $\vec{v_A} = \vec{v_B} + \vec{v_{A/B}}$, where:
     • $\vec{v_B}$: Velocity of reference particle $B$.
     • $\vec{v_{A/B}}$: Velocity of $A$ as observed from the translating frame attached to $B$.
   • Since axes translate without rotation, $\vec{v_{A/B}}$ is the derivative of $\vec{r_{A/B}}$ in the moving frame.

3. Acceleration Relationship:

   • Similarly, $\vec{a_A} = \vec{a_B} + \vec{a_{A/B}}$, where:
     • $\vec{a_B}$: Acceleration of $B$.
     • $\vec{a_{A/B}}$: Acceleration of $A$ measured from $B$’s frame.

Why Translating Frames?

• No Rotational Terms: Because axes remain parallel, angular velocity ($\vec{\omega}$) and angular acceleration ($\vec{\alpha}$) do not appear. This simplifies equations: $$\vec{v_{A/B}} = \dot{\vec{r_{A/B}}}, \quad \vec{a_{A/B}} = \ddot{\vec{r_{A/B}}}$$
• Intuitive Analysis: Imagine a boat ($A$) crossing a river with flowing water ($B$). The boat’s velocity relative to water is $\vec{v_{A/B}}$, while the river’s flow is $\vec{v_B}$.

Problem-Solving Strategy

1. Identify reference particle $B$ and attach a translating frame to it.
2. Express $\vec{v_B}$ and $\vec{a_B}$ in the fixed frame.
3. Compute $\vec{v_{A/B}}$ or $\vec{a_{A/B}}$ using kinematics (e.g., derivatives if $\vec{r_{A/B}}$ is given).
4. Apply: $$\vec{v_A} = \vec{v_B} + \vec{v_{A/B}}, \quad \vec{a_A} = \vec{a_B} + \vec{a_{A/B}}$$

Example

Two cars $A$ and $B$ move on straight roads. Given $\vec{v_B} = 10\ \text{m/s} \ \hat{i}$ and $\vec{v_{A/B}} = -5\ \text{m/s} \ \hat{j}$ (indicating $A$ moves south relative to $B$):

• $\vec{v_A} = 10\hat{i} - 5\hat{j}\ \text{m/s}$.

Common Exam Focus:

• Vector subtraction to find $\vec{r_{A/B}}$, $\vec{v_{A/B}}$, or $\vec{a_{A/B}}$.
• Applications like aircraft in wind or particles on moving platforms.
• Remember: Translating frames only—rotation introduces Coriolis terms (covered later).