91Ó°ÊÓ

When analyzing the motion of two objects moving towards or away from each other, understanding relative speed is crucial. Relative speed is the rate at which the distance between two objects changes. In this case, the red train and the green train are moving towards each other on the same track. To find their relative speed, you add their speeds relative to the ground.

• The speed of the red train is 20 m/s.
• The speed of the green train is 40 m/s.

Therefore, the relative speed of the trains moving towards each other is 60 m/s. This fast closing speed means that the time available for braking before a collision is limited. Thus, calculating whether they will collide involves considering this combined speed and the distance between them.

Braking distance is a measure of how far a moving object travels while coming to a stop. It depends on the object's speed and the rate of deceleration (in this case, braking). For these trains, analyzing braking distance involves using the equation \[ v^2 = u^2 + 2as \]where:

• \( v \) is the final velocity (0 m/s when the train stops),
• \( u \) is the initial speed of the train (20 m/s for the red train and 40 m/s for the green train),
• \( a \) is the deceleration (-1 m/s\(^2\)).

For the red train:

• The stopping distance is 200 m.

For the green train:

• The stopping distance is 800 m.

Together, the required stopping distance for both trains is 1000 meters, which is more than the available 950 meters, confirming that a collision will occur.

In collision analysis, we determine whether two moving bodies will collide based on their trajectories, speeds, and distances. In the given problem, after calculating the individual braking distances of both trains,

• the red train needs 200 m to stop,
• whereas the green train requires 800 m.

When summed up, the necessary stopping distance is 1000 m. However, the trains are only 950 m apart when they begin braking. This fact indicates that the available distance is not enough to prevent a collision. During the analysis, we examine their relative positions and movements, factoring in their speeds and the braking distance. These elements combine to show that kinematics confidently predicts a collision, as limited track length translates into insufficient space for both trains to halt before reaching each other.

To solve problems involving motion, motion equations are indispensable, especially in exercises of kinematics like this. The used equation in our calculations is:\[ v^2 = u^2 + 2as \]This equation is part of the set of kinematic equations, which describes linear motion. The components of this specific equation include:

• \( v \): the final velocity, which can be zero for a complete stop.
• \( u \): the initial velocity.
• \( a \): the acceleration or deceleration.
• \( s \): the distance covered.

For the given trains, respective velocities before stopping are calculated as part of the motion analysis. Here, these equations help bridge the gap between understanding each train's independent behavior and predicting the overall outcome of their encounter on the track. Through accurate calculations, we can assess the critical stopping distances relative to their initial speeds and the combined impact speed if a collision is imminent. These fundamentals of motion underline why a collision is unavoidable when seeking to solve such kinematic questions.