You are driving down a straight road at a speed of \(90 \mathrm{km} / \mathrm{h},\) and you see another car approaching you at a speed of \(110 \mathrm{km} / \mathrm{h}\) along the road. a. Relative to your own frame of reference, how fast is the other car approaching you? b. Relative to the other driver's frame of reference, how fast are you approaching the other driver's car?

Understanding the concept of 'frame of reference' is crucial in solving problems related to relative speed and motion. A frame of reference is basically the viewpoint from which you observe and measure events. Imagine you're standing still and watching a car drive past. Your frame of reference is the ground. Now, if you were in another car driving alongside at the same speed, your frame of reference would be the moving car.
In our exercise, your frame of reference is your car. From your perspective, you are stationary, and everything else is moving around you. When considering the other car approaching, you measure its speed relative to your own car, which simplifies understanding its relative velocity.

Relative velocity refers to the velocity of an object as observed from a specific frame of reference. It is a vector quantity, which means it has both magnitude and direction. The formula for relative velocity of two objects moving towards each other is the sum of their individual velocities when considered from one frame of reference.
In our exercise, we calculate the relative velocity of the approaching car toward your car. Since both cars are moving towards each other, we add the speeds: \(90 \text{ km/h} + 110 \text{ km/h} = 200 \text{ km/h}\). This means that from your viewpoint (frame of reference), the other car's relative velocity is \(200 \text{ km/h}\).
Conversely, from the other driver’s frame of reference, your car is approaching them at the same \(200 \text{ km/h}\), highlighting that relative velocity remains consistent for both parties involved.

Speed calculation becomes straightforward when dealing with relative speeds, especially when the direction of movement is considered. Here, since both cars are on a straight road and moving towards each other, we simply sum their speeds.
Given your speed is \(90 \text{ km/h}\) and the other car's speed is \(110 \text{ km/h}\), you add these values to find the relative speed, yielding: \(90 \text{ km/h} + 110 \text{ km/h} = 200 \text{ km/h}\). This summing method holds because the cars' movements are directly towards each other.
Understanding the conditions under which speeds are added or subtracted is essential in various problems involving relative speed.

The very foundation of this exercise lies in understanding the concept of motion. Motion describes an object’s change in position over time. In our problem, both cars are in motion along a straight road.
When two objects move towards each other, the distance between them decreases over time, which incorporates their individual speeds into a single relative speed. In this scenario, this forms the basis for understanding how quickly the two cars approach each other.
To further grasp this concept, imagine if both cars moved at the same speed: \(100 \text{ km/h}\). From each driver's frame of reference, the other car would still appear to approach at \(200 \text{ km/h}\), showing the essence of relative motion.
By breaking down the problem into steps, we reflect deeply on how motion and speed are interrelated, emphasizing that relative speed is a combined effect of both objects' motion towards one another.