A train of 100 m in length runs parallel to a car with negligible dimensions. Their movements are in a straight line, in the same direction, and the speed of the car is twice the train speed, both constants. What is the distance traveled by car to overtake the train?


Problem data:

• Train length:    d = 100 m;
• Train speed:    v_t = v;
• Car speed:    v_a = 2v.

Problem diagram:

As the car has negligible dimensions compared with the dimensions of the train, it can be considered a point, while the train dimensions are relevant to the problem.
Overtaking begins as the car reaches the back of the train and ends when it reaches the front of the train (Figure 1).

We choose a reference frame pointing to the right. The problem can be reduced to a point, representing the car, at the origin of the reference frame, S_0a = 0, with a speed 2v, and another point, representing the front of the train 100 m ahead S_0t = 100 m, with speed v. Overtaking occurs when these two points are in the same position.

Solution

The two points have a constant speed, equation of displacement as a function of time is given by writing the equations for the two points, for the car for the train Imposing the condition that when both bodies meet, they occupy the same position in the trajectory, equate the expressions (I) and (II) that will be the time that the overtaking takes to happen, substituting this result in any of the equations we have the displacement of the car to overtake the train, substituting into expression (I)