A passenger is late for a train. He stood at a railway station while the accelerating train passed him car by car. The passenger noticed that the last train car passed him for 8 seconds and the previous train car for 10 seconds. How long was the passenger late for the train? You should consider that the train moved with uniform acceleration and the length of the train cars is the same.

###### Solution.

The position-time equation of motion of the train is given by the formula

x_{f} = x_{i} + v_{i}∙t + (a∙t^2)/2

The length of the train cars could be denoted by d. Then the equation for the last train car is

x_{f} – x_{i} = d = v_{1}∙t_{1} + (a∙t_{1}^2)/2 (1)

where v_{1} – the initial velocity of the last train car from the point of view of the passenger and t_{1} – the time it takes the last train car to pass the passenger.

The equation for the previous train car is

d = v_{2}∙t_{2} + (a∙t_{2}^2)/2 (2)

where v_{2} – the initial velocity of the previous train car and t_{2} – the time it takes the previous train car to pass the passenger.

The equation for velocity as a function of time is given by the formula

v_{f} = v_{i} + a∙t

Obviously, that the train was not moving at the beginning, v_{i} = 0. The final velocity v_{f} can become the initial velocity v_{1} of the last train car from the point of view of the passenger and the equation is

v_{f} = v_{1} = a∙(t + t_{2}) (3)

where t – a period of time by which the passenger was late and t_{2} – the time it takes the previous train car to pass the passenger.

The equation for initial velocity v_{2} of the previous train car is

v_{2} = a∙t (4)

Now, we substitute equation (3) into (1) and (4) into (2).

d = a∙(t + t_{2})∙t_{1} + (a∙t_{1}^2)/2 (5)

d = a∙t∙t_{2} + (a∙t_{2}^2)/2 (6)

We equate the right sides of equations (5) and (6) and by dividing by a obtain

(t + t_{2})∙t_{1} + t_{1}^2/2 = t∙t_{2} + t_{2}^2/2

We can write the last equation as

t = (t_{1}∙t_{2} + 0.5∙(t_{1}^2 – t_{2}^2))/(t_{2} – t_{1})

Finally, we can substitute values and calculate

t = (8∙10+0.5∙(8^2 -10^2))/(10-8) = 31 s

If you want to know how to solve such problems. Please try Baby Steps In Physics.

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