A freight train travels from Oslo to Drammen at a velocity of \(50 \mathrm{~km} / \mathrm{h}\). An express train travels from Drammen to Oslo at \(200 \mathrm{~km} / \mathrm{h}\). Assume that the trains leave at the same time. The distance from Oslo to Drammen along the railway track is \(50 \mathrm{~km}\). You can assume the motion to a long a line. (a) When do the trains meet? (b) How far from Oslo do the trains meet?

Relative velocity is a key concept in kinematics. It's essential when understanding how two moving objects come towards or go away from each other. In this exercise, we're dealing with two trains moving towards each other from different points. The freight train travels at 50 km/h from Oslo to Drammen, while the express train speeds at 200 km/h from Drammen to Oslo.
The relative velocity here is the sum of their individual velocities, because they're moving towards each other. This results in a combined relative velocity of \[ 50 \text{ km/h} + 200 \text{ km/h} = 250 \text{ km/h} \].
This simplification helps in finding how fast the gap between the two trains is closing. So, instead of two separate movements, we treat it as a single effective motion with a higher speed. This is much simpler and efficient for solving problems like when and where they meet.

Following a systematic approach when solving problems can make even complex ones seem straightforward. Let's break down the given steps and why each of them is important:

- **Step 1: Define Variables and Equations**
Setting up the stage is crucial. We define variables (\tt\tt for time) and equations to represent the distances both trains cover. \[ d_f = 50t \] represents the distance the freight train travels, and \[ d_e = 200t \] is for the express train.
- **Step 2: Total Distance Equation**
We establish that the sum of both distances must equal the distance between their starting points (50 km). \[ d_f + d_e = 50 \]
- **Step 3: Substitute the Distances**
Replacing symbols with their corresponding expressions simplifies our equation to \[ 50t + 200t = 50 \]
- **Step 4: Solve for Time**
Combining and simplifying terms helps solve for \tt t \tt : \[ t = \frac{50}{250} = 0.2 \text{ hours} \]
- **Step 5: Calculate the Distance from Oslo**
Finally, we find the distance the freight train travels to the meeting point from Oslo: \[ d_f = 50 \times 0.2 = 10 \text{ km} \]
Following these steps methodically ensures no step or concept is missed, making the problem easier to handle.

Understanding the distance-time relationship is fundamental in kinematics problems. Simply put, distance covered is the product of speed and time. This relationship forms the basis of our equations. For the freight train, traveling at a constant speed of 50 km/h, the distance it covers can be expressed as: \[ d_f = 50t \].
Similarly, for the express train moving at 200 km/h, we write: \[ d_e = 200t \].
This relationship simplifies the problem-solving process, enabling us to set up equations that describe the motion. By understanding how time (\tt t \tt ) and distance (\tt d \tt ) correlate, we can predict different aspects of the motion.
In this specific exercise, using the distance-time relationship ensures accuracy in predicting when and where both trains meet. It shows that time can be determined by setting distances traveled by both trains equal to the total distance between Oslo and Drammen. Once \tt t \tt is found, it can be used to figure out any other required distances, as demonstrated in the example.