A teacher and an athlete have a race. Determine the outcome.

Question

Andrew challenges his teacher to have a race with his cousin which is Usain Bolt and he says if he wins he has to give him an A in math. If he doesn't he'll make up all his homework. 15 seconds after the buzzer, the teacher has 135 meters to go to the finish line. 32 seconds after he left the starting line, the teacher only has 50 more meters more to run. 22 seconds after Usain Bolt left the starting line, he is 189 meters from the line. 36 seconds after the buzzer, he is 43 meters from the finish line. Both are running at a constant speed. Determine the outcome of the race.

Answer 1

Mans,

I have to say this looks more like some kind of riddle than a book problem. Any sane person would immediately assume Bolt would win. But what it doesn't tell us is that the teacher was given a leg up.

 

The equation needed is simply this:  velocity = change in distance / change in time,   OR

 

V = Δd/Δt                    Evaluate for each runner.

 

V_T = (135 - 50) / (32 - 15) = 85 / 17 = 5 m/s

 

V_B = (189 - 43) / (36 - 22) = 146 / 14 = 10 3/7 m/s

 

This would clearly support this original assumption of Bolt trouncing the teacher. But when you look at the two racers side by side, you see that at 15 seconds into the race, the teacher is closer to the finish line than Bolt is at 22 seconds into the race. If they are both running a constant speed, how can this be?

 

If we take the calculated velocities and plug them into d = v*t, we find that the teacher's initial 15 seconds only took him 75m, while Bolt's initial 22 seconds took him 229.4m.  If we add each to their respective distances from the finish line, we see that the teacher's course was only 210m while Bolt's was 418.4m. So Bolt's speed was just over double the teacher's speed, his course was just under double the teacher's course. To be more precise, we use the calculated velocities to determine the total time for each.  t = d/v

 

               distance / velocity    = time

Teacher      210m  /   5m/s      = 42 seconds

Bolt         418.4m  /(73/7 m/s) = 40.1 seconds

 

Bolt in deed won, but by a smaller than expected margin.