Lesson Two cars and a train make a race

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Two cars and a train make a race

Problem 1

At a certain place the freeway is directly next to and parallel to the railway line.
Two cars on the freeway are travelling in the same direction as the train. The one car is moving at 1.5 times the speed of the other.
At a certain instant the drivers of both cars are simultaneously directly next to the rearmost point of the train.
The slower car takes 60 seconds to reach the front point of the train. The train itself takes 30 seconds to cover a distance
equal to its own length. If the train and the cars each travel at a constant speed, how long will it take the faster car to reach
the front point of the train?

Solution

Let "r" be the speed of the slower car (in meters per second), 

let  L  be the length of the train (in meters), and

let "v" be the speed of the train (m/s).


The "distance" equation for the slower car is

    60*r = L + 60*v.      (1)


We also are given that  

    L = 30*v;             (2)


hence (substituting (2) into (1) and excluding L)


    60*r = 30v + 60v,   or

    60r  = 90v

    r    =  = .


Then the speed R of the faster car is  

    R = 1.5*r =  = .      (3)


The "distance" equation for the faster car is

    T*R = L + T*v     (4)

where "T" is unknown time under the question.


Substitute  R =   from (3)   and  L = 30v  from (2) into equation (4).  You will get

     = 30v + T*v.


Cancel the factor "v" in both sides.  You will get

     = 30 + T.


Multiply both sides by 4 and simplify to solve for T


    9T = 120 + 4T

    9T - 4T = 120,

    5T      = 120,

     T      =  = 24.


ANSWER.  The unknown time T under the question is 24 seconds.

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