How Can a Train Travel 1km Optimally by Accelerating and Braking?

[philok]

Homework Statement

A commuter travels between two stations. the stations are only 1.00km apart which means the train never reaches maximum crusing speed. The engineer minimizes the time interval (delta t) between the two stations by accelerating for a time interval (delta t sub 1) at a rate (a sub 1 ) = 0.100 m/s^2 and then immediately braking with acceleration (a sub 2) = -0.500 m/s^2 for a time interval (delta t sub 2).
Find the minimum time interval of travel ( delta t) and the time interval (delta t sub 1).

Homework Equations

The only equations i know are the kinematic equations but I am not sure how to apply them to this question

The Attempt at a Solution

 

[Chewy0087]

Come on, what have you tried so far?

Here's a starter, call the distance he covers whilst accelerating $$D_{1}$$, the distance he covers whilst decelerating $$D_{2}$$. You also know the acceleration of the train while accelerating and decelerating. What can you say about $$D_{1} + D_{2}$$ ? Try writing out the equations and see if you can get unknowns to cancel.

 

[kerimek]

I would first gather all the necessary information and data provided in the question. I would then use the kinematic equations to solve for the minimum time interval of travel (delta t) and the time interval (delta t sub 1). The kinematic equations are:

1. x = x0 + v0t + 1/2at^2
2. v = v0 + at
3. v^2 = v0^2 + 2a(x-x0)
4. x-x0 = v0t + 1/2at^2

In this scenario, the initial position (x0) is 0 km, the final position (x) is 1 km, the initial velocity (v0) is 0 m/s, and the final velocity (v) is also 0 m/s. We also know that the acceleration (a) is different during the two time intervals (delta t sub 1 and delta t sub 2).

Using equation 1, we can solve for the minimum time interval of travel (delta t) by plugging in the given values:

1 = 0 + 0(delta t) + 1/2(0.100)(delta t)^2
1 = 0.005(delta t)^2
(delta t)^2 = 200
delta t = √200
delta t = 14.14 seconds

Next, we can use equation 2 to solve for the time interval (delta t sub 1):

0 = 0 + (0.100)(delta t sub 1)
delta t sub 1 = 0 seconds

Therefore, the minimum time interval of travel (delta t) is 14.14 seconds and the time interval (delta t sub 1) is 0 seconds.

I would also check my solution by plugging in the values for the time intervals in equation 4 and making sure that the final position (1 km) is reached. This confirms that our solution is correct.

In conclusion, the engineer can minimize the time interval between the two stations by accelerating for 0 seconds and then braking for 14.14 seconds, resulting in a total time interval of 14.14 seconds.