A high-speed bullet train accelerates and decelerates at the rate of 10 ft/s2. Its maximum cruising speed is 90 mi/h.

Question

A high-speed bullet train accelerates and decelerates at the rate of 10 ft/s^2. Its maximum cruising speed is 90 mi/h.

 

(a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes?

(b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?

(c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart.

(d) The trip from one station to the next takes at minimum 37.5 minutes. How far apart are the stations?

Answer 1

First, make sure that all the units are consistent. If you prefer to work with feet and second, then change mi/h to ft/s,  and minutes to seconds.

a.

a= dv/dt = dv/ds . ds/dt

             = dv/ds . v

 

Rearranging,

a.ds = v dv

Integrating both sides,

a s = 1/2 v²  where the limits of integration for v are 0 and 90mi/h, and a = 10 ft/s².

 

From this, you can find the distance the train traveled from rest up to its cruising speed.

Add to this distance, the distance it traveled for a further 15 minutes.

 

b. Let t = time to accelerate to cruising speed = time to decelerate to stop.

Then a = Δv/ t ⇒ t =Δv/a = v_max /a

Distance traveled during this time = t x average speed = (v_max/a) x v_max /2 = v²_max /2a

Maximum distance traveled = v²_max /2a + (15 min - 2 v _max /a) v_max  +  v²_max /2a

 

c. From part b, the time to reach cruising speed = v_max /a = time to slow down from cruising speed to a stop.  During these times, the train travels a distance of 2 x v²_max /2a = v²_max /a

Cruising time = (45 miles - 2 x v²_max /2a)/ v_max

Minimum time = 2v_max /a + (45 miles - v²_max /a)/ v_max

 

d. Cruising time = 37.5 minutes - 2v_max/a

Distance traveled during cruising time = (37.5 minutes - 2v_max/a) x v_max

Distance between stations = v²_max /a + (37.5 minutes - 2v_max/a) x v_max

 

I leave you to do all the conversion and substitution.