Physics question (complicated)?

Question

A commuter train travels between two downtown stations. Because the stations are only 1.18 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 = -0.360 m/s2 for a time interval t2. Find the minimum time interval of travel t and the time interval t1.

I know this seems tough but i really need help with this problem. Thankyou

Answer 1

the equations are
first from rest to maximum speed
v1=0.100*t1
and d1=.5*0.100*t1^2

from max speed to the next station where it will stop
0=v1-0.360*t2
and
d2=v1*t2-.5*0.360*t2^2

d1+d2=1180 m

t1+t2=t

plugging the v1 into the t2 velocity question

0=0.100*t1-.360*t2
t1=3.60*t2

and
.5*0.100*t1^2+v1*t2-
.5*0.360*t2^2=1180

.5*0.100*(3.60*t2)^2+
0.100*3.60*t2^2-.5*0.360*t2^2=1180

we now have a quadratic to solve for t2
.5*0.100*(3.60*t2)^2+
0.100*3.60*t2^2-.5*0.360*t2^2=1180

t2=sqrt(1180/.828)

t2=37.75 seconds

now solve for t1

t1=135.9 seconds

d1=923.48 m

v1=13.5 m/s

d2=256.5 m

d1 + d2 checks.

t2=37.75 seconds

t1=135.9 seconds

t=173.65

This is the only possible solution based on the question, so by definition it is the minimum.

j