Challenge 2 - Tortoise and Hare
Challenge 2 is all about working out the distance, speed and time of two runners.
Maths teacher Chris Smith and pupils from Grange Academy are here to explain.
The Maths Week Scotland Daily Challenges have been set by the Scottish Mathematical Council.
Chris: This problem is all about distance, speed and time. Ellie and Priya are training at the local athletics club.
There's two athletes, Ellie and Priya, and they're running around a 400 metre track.
Ellie starts at the start, but Priya starts halfway round the track. Ellie - she's running at six metres per second.
But Priya is running at 4.4 metres per second. So my question is, when will Ellie catch Priya? How far around the track will it be? And after how long? Explain your answer.
Pupil: How far in front of Ellie as Priya at the start. How can you use that?
Pupil: Try to think about the difference in their speeds.
Pupil: It might be easier to work out the time first, then the distance.
Chris: What are we trying to work out there? We were seeing that…
Pupil: Priya's got a head start.
Chris: Yeah, and we wanted to know how long it's going to take to make up that lead.
Pupil: Do your best.
So here's the challenge:
Ellie and Priya are training at their local athletics club.
On the 400 metre track, Ellie starts running at the start/finish line.
Priya is exactly halfway round the track when she starts running at exactly the same time.
Ellie runs at 6 metres per second and Priya runs at 4.4 metres per second.
After how long does Ellie catch up with Priya?
Where on the track does this happen?
Need a hint?
- How far in front of Ellie is Priya at the start? How can you use that?
- What can the difference in their speeds tell us?
- It might be easier to work out the time first, then the distance.
Solution
Worked out the answer? Here's how you can do it.
Did you work out how long it took? And where on the track Ellie caught up with Priya?
Since Priya starts halfway around the 400 metre track. She's 200m ahead of Ellie when they start running.
We need to work out how long it takes Ellie to run this extra 200m.
Ellie runs at 6 metres per second and Priya at 4.4 metres per second.
6 take away 4.4 is 1.6, so Ellie catches up by 1.6m every second.
To find the time it takes us to catch up, we divide 200m by 1.6 metres per second to get 125 seconds.
Ellie catches up with Priya after 125 seconds.
To find out where on the track this happens, we have to multiply Ellie's time by her speed.
Ellie catches up with Priya after she's run for 125 seconds at 6 metres per second. So that's equal to 750m.
Since the distance of one lap is 400m, Ellie's run one full lap and then another 350m.
So she catches up with Priya 50m before the finish line.
Well done if you got this one in record time.
Step 1
Since Priya starts halfway round the 400 metre track, she is 200m ahead of Ellie when they start running.
We need to work out how long it takes Ellie to run this extra two hundred metres.
Step 2
Ellie runs at 6 metres per second and Priya at 4.4 metres per second.
6 - 4.4 = 1.6
So Ellie catches up by 1.6 metres every second.
Step 3
To find the time it takes her to catch up, we divide 200 metres by 1.6 metres.
200 ÷ 1.6 = 125
Ellie catches up with Priya after 125 seconds.
Step 4
To find out where on the track this happens, we have to multiply Ellie's time by her speed.
Ellie catches up with Priya after she has run for 125 seconds at 6 m/s.
125 X 6 = 750
Ellie catches up with Priya after running 750m.
Step 5
Since the distance of one lap is 400m, Ellie has run one full lap and then another 350 metres when she catches up with Priya.
So Ellie catches up with Priya 50 metres before the finish line after 125 seconds.
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