Two trains simultaneously left points M and N and headed toward each other. The distance between point M and N i

Question

Two trains simultaneously left points M and N and headed toward each other. The distance between point M and N is 380mi. The speed of the train, which started, from point N was 5 mph faster than the speed of the other train. In two hours after the departure the distance between the trains was 30 mi. What is the speed of each train?

Answer 1

distance=rate*time

d=rt

d=r(2) for the train from point M

d=(r+5)(2) for the train from point N

these 2 distances plus 30 miles equals 380 miles

2r+2(r+5)+30=380

2r+2r+10+30=380

4r+40=380

4r=380-40

4r=340

r=340/4

r=85 mph for one train

r+5=85+5=90 mph for the other train

check:

85*2+90*2+30=380

170+180+30=380

380=380

Answer 2

Lets denote position M as our 0 point. Thus point N is at 380 mi

Let m = speed of train starting from point M. Let x_m be distance train has traveled.

Let n = speed of train starting from point N. Let x_n be distance train has traveled.

 

Since speed = distance/time we have

x_m = 0+m*t (remember we are taking point M as 0 point)

x_n = 380-n*t (point N is at 380

 

We are given speed of train starting at point N was 5 mph faster than one starting at point M. Thus

n = m+5

 

We are told after t = 2 hrs that the distance between the trains was 30 mi. Thus

|x_m-x_n| = |(0+m*t)-(380-n*t)| = |m*t-380+n*t| = |m*t-380+(m+5)*t| = |2m*t+5t-380|

Plug in t = 2

|2m*2+5*2-380| = |4m+10-380|=|40-370| = 30

 

So we have two equations

4m-370 = 30 which means m = 400/4 = 100 mph

or

-(4m-370) = 30 which means 4m-370 = -30 which means m = 340/4 = 85 mph

 

From this n = 105 mph or n = 90 mph

 

Option 1:

If m = 100 mph then x_m = 100*2 = 200 miles

This means n = 105 mph and x_n = 380-105*2 = 170 miles

I think we can discard this because this means the trains have passed each other. I'm thinking the problems wants before they pass each other.

 

Option 2:

If m = 85 mph then x_m = 85*2 = 170 miles
This means n = 90 mph and x_n = 380-90*2 = 200 miles

This is probably the likely case.

 

So your answer is the train leaving from M was at 85 mph and the train leaving from point N was at 90 mph.

 

Hope that helps. Best - David