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A train starts from rest and moves with a constant acceleration for the first 1 km. For the next 3 km, it has a constant velocity and for the last 2 km, it moves with constant retardation to come to rest after a total time of motion of 10 min. Find the maximum velocity and the three time intervals in the three types of motion.

A train starts from rest and moves with a constant acceleration for the first 1 km. For the next 3 km, it has a constant velocity and for the last 2 km, it moves with constant retardation to come to rest after a total time of motion of 10 min. Find the maximum velocity and the three time intervals in the three types of motion.

Grade:11

1 Answers

Shaswata Biswas
132 Points
7 years ago
Suppose the velocity of the train increases from 0 to v in time t1, then moves with constant speed v upto time t2 and finally its velocity decreases from v to 0 between time interval t2 and t.
For motion between t=0 and t=t1 let acceleration be \alpha.
Then, \alpha = (v-0)/(t1-0)  = v/t1
=> t1 = v/\alpha
Similarly, for retardation \beta
\beta = (v-0)/(t-t2)  = v/(t-t2)
=> t-t2 = v/\beta
=> t1+t-t2 = v/\beta + v/\alpha
=> t2-t1 = t - v{1/\alpha + 1/\beta}
Draw a graph of the motion of the train, from where we can find the displacement of the train.
Now, for total displacement S = 1+3+2 = 6 km, and total time taken t=10 min = 1/6 h, we have.
S = v.t1/2 + v(t2-t1) + v(t-t2)/2
S = v2/2\alpha + v[t- v{1/\alpha + 1/\beta}] + v2/2\beta
S/v = v{1/\alpha + 1/\beta}/2 + t - v{1/\alpha + 1/\beta} = t - v{1/\alpha + 1/\beta}/2
or, t = S/v + v{1/\alpha + 1/\beta}/2
Puutting the values of the variables, the max. velocity attained "v" can be claculated.
THANKS

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