Final answer:
The question requires calculating the relative speed of a man running to a moving train, as observed from inside the train. Using kinematics and vector addition, the relative speed is found by either taking the train's speed alone or computing vector components followed by their resultant using the Pythagorean theorem.
Step-by-step explanation:
The question involves determining the relative velocity of two objects, a man running and a train moving, with respect to an observer seated inside the train. Using principles from kinematics and vector analysis, we break down the velocities into components along and perpendicular to the track, and then use the Pythagorean theorem to find the resultant velocity or speed that the observer will notice.
Relative Speed Perpendicular to Track
When the man runs perpendicular to the train track at 12 m/s and the train moves at 30 m/s, the speed of the man relative to a passenger in the train is simply the train's speed, which is 30 m/s, because their motions are perpendicular and the man's velocity does not contribute to his speed relative to the passenger in the direction of the train's motion.
Relative Speed at an Angle to the Track
For the case where the man runs at an angle of 30° to the track, we calculate the component of the man's velocity parallel to the track by multiplying 12 m/s by cos(30°), and then find the relative speed by subtracting this component from the train's speed. Next, we find the perpendicular component of the man's velocity using 12 m/s times sin(30°). The final step involves adding these two components using the Pythagorean theorem to find the resultant velocity, which will be the speed of the man relative to the passenger in the train.
Conclusion
The techniques used here are core principles of physics, illustrating how to analyze and solve problems involving relative motion, vector addition, and reference frames.