Welcome to our exploration of relative motion in one dimension. Motion is always described relative to a reference frame, and the same motion can appear completely different when viewed from different perspectives. Let's consider a person walking on a moving train to illustrate this fundamental concept.
From the ground observer's perspective, both the train and the person are moving to the right. However, from the train observer's perspective, the person appears to be moving much slower, or might even appear stationary if they're walking at the same speed as the train.
A reference frame is a coordinate system used to describe the position and motion of objects. It consists of an origin point and coordinate axes that allow us to measure positions, velocities, and accelerations.
Let's place an object in our coordinate system. From the ground frame, we can measure its position using a position vector. However, if we have a moving reference frame, the same object will have a different position vector in that frame.
The fundamental equation for relative velocity in one dimension is v_AB equals v_A minus v_B. This tells us the velocity of object A as observed from the reference frame of object B.
Let's derive this formula step by step. Starting with position, the relative position x_AB equals x_A minus x_B. Taking the time derivative of both sides gives us the relative velocity formula.
Now let's see this formula in action. Here we have two objects on a number line. Object A moves with velocity +3 units per second, and object B moves with velocity +1 unit per second. The relative velocity of A with respect to B is 3 minus 1, which equals +2 units per second.
Let's work through three practical examples to master the relative velocity formula. Each example demonstrates a different scenario you might encounter in physics problems.
Example 1: Two cars moving in the same direction. Car A travels at 15 meters per second, car B at 10 meters per second. The relative velocity is 15 minus 10, equals positive 5 meters per second. This means car A is moving 5 meters per second faster than car B.
Example 2: Cars moving in opposite directions. Car A moves at positive 20 meters per second, car B at negative 15 meters per second. The calculation becomes 20 minus negative 15, which equals positive 35 meters per second. The relative velocity is much larger when objects move in opposite directions.
Example 3: One object stationary. Car A moves at 12 meters per second while car B is at rest. The relative velocity is simply 12 minus 0, equals 12 meters per second. When one object is stationary, the relative velocity equals the velocity of the moving object.
Now let's explore advanced applications of relative motion. These scenarios demonstrate how relative velocity concepts apply to real-world situations like collisions, overtaking, and multi-reference frame problems.
In collision analysis, two cars approach each other. Car A moves at positive 25 meters per second, car B at negative 20 meters per second. Their relative approach speed is 45 meters per second, which determines the severity of impact.
For overtaking scenarios, a fast car traveling at 30 meters per second overtakes a slower car at 20 meters per second. The relative velocity of 10 meters per second determines how quickly the overtaking occurs.
In elevator motion, we deal with multiple reference frames. A person moving upward in an elevator that's also moving upward has a ground velocity equal to the sum of both velocities. Remember to define your reference frames clearly and apply the formulas consistently.