Two cars run a straight road with constant speeds v₂>v₁, the two cars run with a time interval T and from points separated by a distance D over the road. Assuming that car 1 starts motion before car 2, determine after how long, after car 2 starts, they will meet, assuming they move:
a) In the opposite directions;
b) In the same direction, from the position of car 2 to car 1.


Problem data:

• Car speed 1:    v₁;
• Car speed 2:    v₂;
• Time interval between cars:    T;
• Distance between the starting points of the two cars:    D.

Solution

a) We choose a reference frame pointing to the right. Car 1 starts of the origin, S₀₁ = 0, in the direction of reference frame with speed v₁, and car 2 starts from a point at a distance D of the first car, S₀₂ = D, in the opposite direction, and its speed will be −v₂ (Figure 1).


Car 2 starts in an instant t and, as car 1 starts an instant t before car 2, when car 2 stars, car 1 is already on the move, there is a time equal to (t+T).
The equation for displacement as a function of time with constant velocity is given by dado por writing the expression (I) for each car, for car 1 for car 2 When the two cars meet, they are in the same position on the road, equating the expressions (II) and (III) factoring time t on the left-hand side
b) We choose the same reference frame as the previous item. Car 1 starts from the origin S₀₁ = 0, in the opposite direction of the reference frame with speed −v₁. Car 2 starts from a point at a distance D from the first car S₀₂ = D, also in the opposite direction of the reference frame, and its speed will be −v₂ (Figure 2).


Writing the expression (I) for each car, for car 1 for car 2 When the two cars meet, they are in the same position on the road, equating expressions (IV) and (V) factoring time t on the left-hand side