Relative Motion Applications

The Core Principle of Relative Motion

At its heart, relative motion is a strategy for simplification. By switching our frame of reference to one of the moving objects, a complex problem involving two or more moving bodies can often be reduced to a simpler problem of a single body moving relative to a stationary point.

The fundamental tool is the vector subtraction formula for relative velocity: the velocity of object A relative to object B (${\vec{v}}_{AB}$) is given by:

Here, ${\vec{v}}_A$ and ${\vec{v}}_B$ are the velocities of A and B with respect to a common stationary frame (usually the ground). A common error is to add the vectors instead of subtracting them; precision in vector subtraction is non-negotiable.

For most 2D problems, the most reliable method is to break down all velocities into components (e.g., $\hat{i}$ and $\hat{j}$) and apply the subtraction component-wise:

Let's begin by reviewing the foundational concepts and problem classifications.

Application 1: River-Crossing and Wind-Aeroplane Problems

This is a classic problem archetype in which an object moves in a medium that is itself moving. The key is to correctly combine the velocity of the object relative to the medium and the velocity of the medium relative to the ground.

Let's define the vectors:

• ${\vec{v}}_{b,w}$: Velocity of the boat with respect to water (the direction the boat is steered).
• ${\vec{v}}_{w,g}$: Velocity of the water with respect to the ground (the river current).
• ${\vec{v}}_{b,g}$: Velocity of the boat with respect to the ground (the boat's actual path).

The relationship is a simple vector addition:

This relationship is visually broken down in the following diagram.

Almost all problems in this category boil down to two specific objectives:

1. Crossing in Shortest Time: To cross the river of width 'd' as quickly as possible, the component of velocity perpendicular to the banks must be maximized. This is achieved by pointing the boat directly across the river (${\vec{v}}_{b,w}$ is perpendicular to ${\vec{v}}_{w,g}$). The current will carry the boat downstream, resulting in a "drift," but the crossing time $t=d/v_{b,w}$ will be minimal.

2. Crossing via Shortest Path (Minimum Drift): To reach a point directly opposite the starting position (zero drift), the boat must be steered upstream at an angle such that the component of its velocity relative to the water cancels out the river current. The resultant velocity ${\vec{v}}_{b,g}$ will be directed straight across. This is only possible if $v_{b,w}>v_{w,g}$.

The physics of wind-aeroplane problems is identical. The aeroplane is the boat, and the wind is the river.

Application 2: Multi-Body Pursuit and Closest Approach

This second major category involves objects moving independently, where we need to determine if, when, or where they collide, or what the minimum separation between them is.

The master technique is to analyze the motion from the reference frame of one of the objects. For two particles A and B, if we work in the frame of B, then B is stationary at the origin. Particle A's motion is then described by the relative velocity ${\vec{v}}_{AB}$. The problem is now reduced to analyzing the path of a single particle 'A' relative to a fixed point 'B'.

Collision

For two particles A and B to collide, the relative velocity vector ${\vec{v}}_{AB}$ must be directed along the line joining them, i.e., along the initial relative position vector ${\vec{r}}_{BA}={\vec{r}}_A-{\vec{r}}_B$. If ${\vec{v}}_{AB}$ has any component perpendicular to ${\vec{r}}_{BA}$, the particles will miss each other.

A common JEE Advanced variant is the "pursuit curve," where multiple particles at the vertices of a regular polygon move towards the next particle in sequence.

Closest Approach / Minimum Distance

If the particles do not collide, we are often asked to find the minimum separation between them. This occurs precisely when their relative velocity is perpendicular to the line joining them. At this point, the velocity of approach becomes zero. There are two primary methods to solve this:

1. Calculus Method: Express the square of the distance between the particles, $D^2(t)$, as a function of time. The minimum distance occurs when $d(D^2)/dt=0$. This method is robust and always works, especially when accelerations are involved.

2. Relative Velocity Method: In the reference frame of particle B, particle A moves along a straight line with velocity ${\vec{v}}_{AB}$. The minimum distance is then the perpendicular distance from point B to this line of motion. This geometric approach can be much faster if the geometry is simple.

Advanced Techniques and Problem Variations

To achieve mastery, you must be exposed to non-standard problems and alternative solution techniques. Standard component analysis is the workhorse, but sometimes a purely geometric or a more advanced calculus approach is more elegant or necessary.

1. Geometric Vector Triangle Method

For some problems, constructing a vector triangle (${\vec{v}}_{b,g}={\vec{v}}_{b,w}+{\vec{v}}_{w,g}$) and applying geometric laws like the Sine Rule or Cosine Rule can be much faster than resolving into components. This is especially true when angles are the primary unknowns.

2. Pursuit Curves with Calculus

In some pursuit problems, the direction of one object's velocity is continuously changing to "aim" at the other. A classic example is a boat always heading towards a point on the shore while being drifted by a current. Solving for the trajectory requires setting up and integrating a differential equation, often in polar coordinates.

Conclusion

In this lesson, we have structured the topic of relative velocity around key problem archetypes crucial for JEE Advanced.

Key Takeaways:

• Fundamental Tool: Relative velocity simplifies multi-body problems. The core equation is ${\vec{v}}_{AB}={\vec{v}}_A-{\vec{v}}_B$, best handled with vector components.
• Problem Categories: We focused on two main types:
  1. River/Wind Problems: Solved by the vector sum ${\vec{v}}_{\text{resultant}}={\vec{v}}_{\text{relative to medium}}+{\vec{v}}_{\text{medium}}$. Key tasks are finding the shortest time or shortest path.
  2. Pursuit/Closest Approach Problems: Solved by analyzing motion in one object's frame. Key tasks are checking for collision or finding minimum separation using either calculus or geometric methods.
• Advanced Techniques: Beyond standard component analysis, purely geometric vector methods and calculus-based trajectory analysis represent a higher level of problem-solving.

With this foundation in advanced kinematics, you are now prepared to move into dynamics. In the next lesson, we will tackle a cornerstone of JEE mechanics: deriving and applying constraint relations in complex systems involving pulleys, wedges, and strings. This will build directly on our use of vectors and reference frames.