A commuter train travels between two downtown stations. Because the stations are

Question

A commuter train travels between two downtown stations. Because the stations are only 1.04 km apart, the train never reaches its maximum possible cruising speed. During rush hour the engineer minimizes the travel interval t between the two stations by accelerating for a time interval t1 at a1 = 0.100 m/s2 and then immediately braking with acceleration a2 = -0.510 m/s2 for a time interval t2. Find the minimum time interval of travel t and the time interval t1. Hint: It is much easier to do this problem graphically. First draw an acceleration versus time graph. What does the area under this graph physically mean? You should be able to get a relation between t1 and t2. Then draw a velocity versus time graph. What does the area under this graph physically mean? You should be able to get another relation between the two time intervals. Now you have two equations and two unknowns.

Draw position, velocity, and acceleration graphs.

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please give answer and steps. thank you

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Explanation / Answer

The maximum speed reached by the train is:
vmax = a1 * t1
and because it must come to rest at the second station,

we also know that:
0 = vmax + a2 * t2 = a1 * t1 + a2 * t2
so
t2 = -(a1/a2)*t1
Now the total distance covered is d=1.04 km
d = 0.5*a1*(t1)^2 + (v_max*t2) + 0.5*a2*(t2)^2
The first term in the above equation is the distance covered while the train is accelerating at a1, and the next two terms are the distance covered while the train is decellerating at a2.
the expressions for Vmax and t2 as functions of t1,

we get:
d = 0.5*a1*(t1)^2 + a1*t1*(-(a1/a2)*t1) + 0.5*a2*(-(a1/a2)*t1)^2
d = 0.5*a1*(t1)^2 - ((a1^2)/a2)*t1^2 + 0.5*((a1^2)/a2)*(t1)^2
d = 0.5*a1*(t1)^2 - 0.5*((a1^2)/a2)*t1^2
d = 0.5*(a1 - (a1^2)/a2)*t1^2
d = 0.5*((a1*a2 - (a1^2))/a2)*t1^2
t1^2 = (2*d*a2)/(a1*a2 - a1^2)

           =(2*1.04 km*(-0.510 m/s2))/((0.1 m/s2)(-0.510 m/s2)-(0.1 m/s2)^2)
t1^2 = 25873.17 sec^2
t1 = 160.85 sec
from equation,

t2 = -(a1/a2)*t1
t2 = (0.1 m/s2/0.51 m/s2)*(160.85 sec) = 31.539 sec
So the total time is
t = t1 + t2

    = 160.85 sec + 31.539 sec=192.389 sec

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