Acommuter train travels between two downtown stations. because the stations are only 1.48 km apart, the train never reaches its maximum possible cruising speed. during rush hour the engineer minimizes the travel interval δt between the two stations by accelerating for a time interval δt1 at a1 = 0.100 m/s2 and then immediately braking with acceleration a2 = -0.510 m/s2 for a time interval δt2. find the minimum time interval of travel δt and the time interval δt1.

Δt=6.43h

Δt1 = 5.38h

Explanation:

We know that the train has to stop at the destination, so in the interval Δt2:

Vf = Vo + a2*Δt2    where Vo is the final velocity in the intervat Δt1.

Vf = a1*Δt1 + a2*Δt2 = 0    Solving for Δt2:

   Let this be eq1.

The total distance is 1.48km, and this is the sum of the displacements during Δt1 and Δt2:

Replacing our previous value for Δt2 in this equation we can solve for Δt1:

Δt1 = 19351.36s ≅ 5.38h    Using this value in eq1, we get Δt2:

Δt2 = 3794.39s ≅ 1.054h    Therefore, total time Δt= 23145.75s ≅ 6.43h

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.