Calculus Problem

Question

A high-speed bullet train accelerates and decelerates at the rate of 10 ft/s^2 . Its maximum cruising speed is 120 mi/h . (Round your answers to three decimal places.) (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 60 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

The statement of the problem uses feet, miles, hours, minutes, seconds.

There is a danger of confusing units, so I will convert everything to

feet and seconds.

Let

a = 10 ft/sec² be the train's acceleration,

V = 120 mi/h = 120 × 5280 ft / 3600 sec = 176 ft/sec be the maximum speed.

In general, the train starting from rest will accelerate, then cruise, then it may decelerate to rest.

Let's use the following symbols:

t is the time the train accelerates,

s is the distance the train covers during acceleration,

T is the time the train cruises,

S is the distance covered while cruising.

This question is in a calculus class, and I do not know whether you are

expected to know the relationships among the above quantities, or

you are supposed to derive them from

the definition of speed, as the integral of acceleration, and

the definition of distance covered as the integral of speed.

In any case, let me just write down the relations, as we will use them below.

V = at

s = at²/2

S = VT

First let's calculate how long it takes the train to accelerate to maximal speed V:

t = V/a = 176/10 = 17.6 sec

During that time the train covers distance

s = at²/2 = 10×17.6²/2 = 1548.8 ft

Notice that the time t and distance s are much smaller than the times and distances

involved in each of the four questions.

That means that in each question the train has enough time to accelerate to its maximal speed V.

So each question then reduces to determining the time T or distance S travelled while cruising.

Question (a):

We are given

T = 15 min = 900 sec

We are to determine

s + S =

s + VT =

1548.8 + 176×900 = 159948.8 ft = 30.3 mi

Question (b):

In this situation the train accelerates for time t,

cruises an unknown time T, and

decelerates for time t.

We are given

t + T + t = 15 min = 900 sec

We are to determine

s + S + s =

2s + VT =

2s + V(900 - 2t) =

2×1548.8 + 176(900 - 2×17.6) = 155302.4 ft = 29.4 mi

Question (c)

Like question (b) the train first accelerates then cruises then decelerates.

We are given

s + S + s = 60 mi = 316800 ft

We are to calculate

t + T + t =

2t + S/V =

2t + (316800 - 2s)/176 =

2×17.6 + (316800 - 2×1548.8)/176 = 1817.6 sec = 30.3 min

Question (d)

This is like question (b) in that we are given time and are to determine distance.

We are given

t + T + t = 37.5 min = 2250 sec

We are to determine

s + S + s =

2s + VT =

2s + V(2250 - 2t) =

2×1548.8 + 176(2250 - 2×17.6) = 392902.4 ft = 74.4 mi

So the stations are at least 74.4 miles apart.

Answer 2

Hi Sizana,

A problem very similar to this one has already been answered on the forum before. It's the same steps but with just different numbers. You can find the problem here: https://www.wyzant.com/resources/answers/174649/a_high_speed_bullet_train_accelerates_and_decelerates_at_the_rate_of_10_ft_s2_its_maximum_cruising_speed_is_90_mi_h

Hope this helps! :)