Trains and relative speed

Train prob­lems use the same for­mula as before: speed = dis­tance ÷ time. The only new thing is the dis­tance. A train is long, so when it "passes" some­thing, it has to travel more than you might think. This les­son shows you how far a train goes when it passes a pole or a plat­form, how to han­dle two mov­ing things at once, and a few other jour­ney prob­lems.

A train pass­ing a pole#

When a train of length $l$ passes a pole, a sig­nal post or a stand­ing man, it trav­els a dis­tance equal to its own length, $l$.

Pic­ture it. The front of the train reaches the pole. The train has passed the pole only when its back end reaches the pole. In that time the front has moved for­ward one full train length.

A train pass­ing a plat­form or a bridge#

When a train passes a plat­form, a bridge or a tun­nel, it trav­els a dis­tance equal to the length of the train plus the length of the plat­form.

The front of the train reaches the start of the plat­form. The train has passed it only when its back end leaves the far end of the plat­form. So the front trav­els the plat­for­m's length and then one more train length.

Rel­a­tive speed#

When two things are both mov­ing, what mat­ters is how fast the gap between them changes. This is called their rel­a­tive speed.

1. If two bod­ies move in the same direc­tion at $x$ km/h and $y$ km/h, with $x$ big­ger than $y$, their rel­a­tive speed is $(x - y)$ km/h.
2. If two bod­ies move in oppo­site direc­tions at $x$ km/h and $y$ km/h, their rel­a­tive speed is $(x + y)$ km/h.

In the same direc­tion, the faster one gains only the dif­fer­ence of the speeds on the slower one. Com­ing towards each other, both close the gap, so the speeds add.

Worked exam­ples#

Exam­ple 1#

A train 360 m long passes a sig­nal post in 18 sec­onds. Find its speed in km/h.

The dis­tance is the length of the train, 360 m.

$$\displaystyle \text{speed} = \frac{360}{18} = 20 \text{ m/s}$$

$$\displaystyle 20 \times \frac{18}{5} = 72, \text{ so the speed is } 72 \text{ km/h}$$

Exam­ple 2#

A train 200 m long passes an elec­tric pole in 10 sec­onds. Find its speed in km/h.

$$\displaystyle \text{speed} = \frac{200}{10} = 20 \text{ m/s} = 20 \times \frac{18}{5} \text{ km/h} = 72 \text{ km/h}$$

Exam­ple 3#

A man walks across a bridge 600 m long in 5 min­utes. Find his speed in km/h.

A man is very short com­pared with a bridge, so we take the dis­tance as just the bridge, 600 m.

$$\displaystyle \text{speed} = \frac{600 \text{ m}}{5 \text{ min}} = 120 \text{ m per minute}$$

There are 60 sec­onds in a minute, so this is $120 \div 60 = 2$ m/s.

$$\displaystyle 2 \times \frac{18}{5} = \frac{36}{5} = 7.2, \text{ so the speed is } 7.2 \text{ km/h}$$

Exam­ple 4#

A train 90 m long is run­ning at 54 km/h. How long does it take to pass a sig­nal pole?

Change the speed into m/s, because the length is in metres:

$$\displaystyle 54 \times \frac{5}{18} = 15 \text{ m/s}$$

$$\displaystyle \text{time} = \frac{90}{15} = 6 \text{ seconds}$$

Exam­ple 5#

A train 150 m long is run­ning at 54 km/h. How long does it take to pass a plat­form 90 m long?

Dis­tance $= 150 + 90 = 240$ m. Speed $= 15$ m/s, as in Exam­ple 4.

$$\displaystyle \text{time} = \frac{240}{15} = 16 \text{ seconds}$$

Exam­ple 6: Two trains, oppo­site direc­tions#

Two trains, 120 m and 80 m long, run on par­al­lel tracks towards each other at 50 km/h and 40 km/h. How long do they take to pass each other com­pletely?

They move in oppo­site direc­tions, so the rel­a­tive speed is $50 + 40 = 90$ km/h.

$$\displaystyle 90 \times \frac{5}{18} = 25 \text{ m/s}$$

To pass each other com­pletely, together they must cover both lengths: $120 + 80 = 200$ m.

$$\displaystyle \text{time} = \frac{200}{25} = 8 \text{ seconds}$$

Exam­ple 7: Two trains, same direc­tion#

The same two trains run in the same direc­tion, the faster one behind. How long does the faster train take to pass the slower one?

Rel­a­tive speed $= 50 - 40 = 10$ km/h, which is $\displaystyle 10 \times \frac{5}{18} = \frac{25}{9}$ m/s.

$$\displaystyle \text{time} = 200 \div \frac{25}{9} = 200 \times \frac{9}{25} = 72 \text{ seconds}$$

Pass­ing in the same direc­tion takes much longer.

Other jour­ney prob­lems#

Exam­ple 8: Part of a jour­ney#

A bus leaves town A for town B at 50 km/h. It cov­ers one third of the dis­tance in 6 hours. Find the dis­tance between A and B.

In 6 hours at 50 km/h, the bus cov­ers $50 \times 6 = 300$ km.

This is one third of the whole dis­tance $x$:

$$\displaystyle \frac{x}{3} = 300 \quad \text{so} \quad x = 900$$

The dis­tance from A to B is 900 km.

Exam­ple 9: What time will it arrive?#

A train leaves New Delhi at 10:50 a.m. at 80 km/h. At what time will it have cov­ered 120 km?

$$\displaystyle \text{time} = \frac{120}{80} = \frac{3}{2} \text{ hours} = 1 \text{ hour } 30 \text{ minutes}$$

Add this to 10:50. 10:50 plus 1 hour is 11:50, and 30 min­utes more is 12:20. The train will have cov­ered 120 km at 12:20 p.m.

Steps for train prob­lems#

1. Decide the dis­tance: the train's length alone, or the train's length plus the other length.
2. If two things move, find the rel­a­tive speed: sub­tract for the same direc­tion, add for oppo­site direc­tions.
3. Put lengths and speeds in match­ing units. With metres, use m/s.
4. Use speed = dis­tance ÷ time.

Prac­tice#

1. A train 240 m long passes a pole in 16 sec­onds. Find its speed in km/h.
2. A train 150 m long runs at 72 km/h. How long does it take to cross a bridge 250 m long?
3. A train 200 m long runs at 90 km/h and passes a plat­form in 20 sec­onds. How long is the plat­form?
4. A man walks 1.2 km in 15 min­utes. Find his speed in km/h.
5. Two trains 100 m and 120 m long run towards each other at 42 km/h and 30 km/h. How long do they take to pass each other?
6. The same two trains run in the same direc­tion at 42 km/h and 30 km/h. How long does the faster one take to pass the slower one?
7. A car leaves at 9:40 a.m. and trav­els at 60 km/h. At what time will it have cov­ered 150 km?
8. A bus at 45 km/h cov­ers two fifths of a jour­ney in 4 hours. Find the length of the whole jour­ney.

Answers#