Question of speed...?

Question

Once, I was on a bike and another person was on a scooter. We started from the same place at the same time. Now, I reached the destination, which was 7 km away, 10 minutes earlier than him. Now the confusing thing is that if I do {7/(10/60)}, it come out that the difference in speeds comes out to be 42 km/hr, which is impossible as my max. speed was 45.

Can anyone explain me the paradox??

Answer 1

You're being sloppy with that 7/(10/60) jazz.

Let
Sb bike speed
Ss scotter speed.

They both covered 7 km, with the scooter taking 10 minutes longer.

Sb=7/t
Ss = 7/(t+10/60)

The difference in speed is
Sb-Ss = 7/t - 7/(t+1/6) = 7(1/t - 1/(t+1/6))

So, in order to get the difference in speeds you need to know the times. Think about it: If it took you 20 hours to cover the 7 km then the scooter would be going just slightly slower to take only 10 minutes more. However, the faster you covered the distance the greater the speed difference would have to be.

Answer 2

Well, as you can see by now, there is no paradox. You are just doing a calculation that has no bearing on the problem.

What does have a bearing is that we do not have quite enough information for a single value answer. In other words, we cannot say, from the information presented, that you went X km/hr and the scooter rider went Z km/hr.

We can, however, give a line on which you can graph corresponding speeds, or better yet, the equation with which you can feed in a speed of interest for one of you and find the speed of the other. AND, to make it better, telling us you never went over 45 km/hr lets us say what the absolutely highest speed the scooter reached as well. Well, not so fast... since the scooter speed could vary over the course length, its maximum speed could have been very high so long as it went slow enough over the rest of the course to make it take 10 minutes longer than you too. Kind of like the rabbit's speed profile in the famous race with the turtle.

But, back to the equation I mentioned. By now, it should have occurred to you that the last point holds for all speed combinations and only by thinking of average speeds over the course can we actually compare speeds. For exactness, the only thing you can truly say with certainty is that you took 10 minutes less than the scooter. But... on to average speeds.

If your speed was X km/hr, then you took (7 km)*(60 min/hr)/(X km/hr) = 420/X minutes. The scooter took (420/X + 10) minutes. If his speed was Z, then he took (7 km)*(60 min/hr)/(Z km/hr) = 420/Z min. So 420/X + 10 = 420/Z. Rearranging gives us: Z = 42X/(42 + X). So now you can feed in any (average) speed you like for yourself and get a corresponding (average) speed for the scooter. For instance, if you had gone 45 km/hr for the entire course (instant acceleration I guess!), then his average speed over the course would be 19.3333 km/hr. More exactly, you would have taken 9.3333 minutes to complete the course and the scooter about 21.72 minutes. To get the exact 10 minute difference, you must have averaged 40.0233 or so km/hr and he 20.4939 km/hr.

So I'll go with that as being the best point on the line. Bike at 40 km/hr and scooter at 20½ km/hr. It is the nearest any of us can point you. Still, it seems unlikely you stayed so near your maximum for so long, but hey, I haven't ridden a bike in 29 years so what do I know, eh?

Answer 3

There's no paradox, since the difference in speeds is not 7/(1/6) = 42.

Speed is distance divided by time. Time is distance divided by speed. Let the greater speed be y, slower speed be x. Distance is 7 km, and difference in times is 1/6 hr, so

7/x - 7/y = 1/6
(7y - 7x)/xy = 1/6
7y - 7x = xy/6
y - x = xy/42

y - x is the difference in speeds, and it's not 42. If you solve that last equation for y,

42y - 42x = xy
42y - xy = 42x
(42-x)y = 42x
y = 42x / (42-x)

No matter what speed you pick for x, I can pick a speed y that will get me there 7 km away 10 min sooner, provided x < 42 km/hr.

Answer 4

You don't work it out like this. To calculate the difference in speeds you need to know his time for the journey. If I use t for his time in minutes then his speed was 7*60/t and your speed was 7*60/(t - 10) then you can work out the difference.
Suppose he took 25 minutes and you took 15 minutes.
His speed would be 420/25 = 16.8 kph and yours would be 420/15 = 28 kph.