A distance and time query

Question

A freight train leaves the station at 9am. an amtrak train leaves the same station 6 hours later at 3pm. the freight train travels 70mph slower than the amtrak train. if the amtrak passes the freight train at 8pm what is the seed of each

Answer 1

Jules,

You need to write an equation for the distance that each train has traveled.

The key is to make sure you are using the same time frame for each.

The basic equation is d = v*t, where d is distance, v is velocity, and t is time.

Since we know their distance is the same at 8 PM, we know the amtrak

has been traveling for 8-3 = 5 hours, while the freight train has been

traveling for an additional 6 or 5+6 = 11 hours.

 

So we can write the following equations:

 

d_f = v_f * 11        and           d_a = v_a * 5

 

And since we know that their distances are the same, d_f = d_a and we can write:

 

v_f * 11 = v_a * 5          but with two variables, we need another equation.

 

So we use what we know about the two velocities, namely that the amtrak is

traveling 70 mph faster than the freight. So we write:

 

v_a = v_f + 70              Now we substitute into the first equation:

 

v_f * 11 = (v_f + 70) * 5      Now solve for v_f

 

11v_f  = 5v_f + 350

 

6v_f = 350

 

v_f = 350/6 = 58.33 mph

 

v_a = 58.33 + 70 = 128.33 mph

 

I hope this helps.

Answer 2

Hi Jules:

 

In both cases, since it is implied the speeds are constant, we can use the basic relationship between distance (d), speed (v), and time (t) for motion at constant speed:

 

d = vt

 

We know that the speed of the freight train, v_fr is 70 mph slower than the speed of the Amtrak train (v_A).  We can express this mathematically as:

 

v_fr = v_A - 70

 

At the time they pass, we know three other things related to this equation:  that the trains have traveled the same distance (d_fr = d_A), that the freight train has been traveling for 11 hours (9 am to 8 pm), and that the Amtrak train has been traveling for 5 hours (3 pm to 8 pm).

 

Since d_fr = d_A, we also know that v_frt_fr = v_At_A (since d = vt in each case).  And, as mentioned in the last paragraph, we know t_fr (the time the freight train has been traveling, 11 hours), and t_A (the time the Amtrak train has been traveling, 5 hours).  Thus:

 

v_frt_fr = v_At_A becomes  v_fr(11) = v_A(5)

 

and given the relationship above between v_fr and v_A, we can further rewrite this last relationship as:

 

(v_A-70)(11) = v_A(5)

 

This is now one equation with one unknown, which you should be able to solve for v_A.  Once you know that, you can calculate v_fr by the speed relationship, and that will give you the complete answer the problem asks for.

 

If you want to check a solution or have any other problems, just let me know.  Good luck!

Answer 3

The two trains travel the same distance.

r represents rate of amtrack

r - 70 represents rate of freight

 

freight: d = (70 - r)(11)

amtrack: d = (r)(5)

 

(r - 70)(11) = (r)(5)

11r - 770 = 5r

-770 = -6r

128 1/3 = r