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Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

### There and Back

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

### Escalator

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

# How Do You React?

### Why do this problem?

This problem brings together several mathematical ideas: modelling, using quadratic graphs to solve problems, rearranging quadratic formulas, and interpreting results in the context of a real-world problem.

### Possible approach

This problem follows on from Reaction Timer. Perhaps spend some time working on the second experiment before working on this problem.

Begin the lesson with this question on the board:

If two people caught the ruler at 15 and 30cm do you think the first person's reactions are twice as fast as the second person's?

Give learners a chance first to think about it on their own, and then to agree an answer with their partner, before discussing it with the rest of the class.

Then pose the question:

What is the relationship between the distance travelled by the ruler and the reaction time?

Allow a short time for learners to speculate about what sort of relationship they might expect to discover. Then hand out this worksheet (Word, pdf) for learners to work on the problem in pairs.
If necessary, bring the class together to talk about units of acceleration and velocity and explain the notation $ms^{-1}$ and $ms^{-2}$, and to clarify any other new ideas met in this task.

Towards the end, give the class time to carry out the experiment and see how their reaction times compare with the average.

### Key questions

As an object falls, when is it travelling slowest/fastest?
What does the graph of distance against time tell you about the change in distance when the time doubles?
Do you think the modelling assumptions are reasonable?

### Possible extension

The Stage 5 problem Cannon Balls explores vertical motion of a cannon ball and solving equations of vertical motion.

When considering the validity of the modelling assumptions, learners could sketch how they think the graph might be changed if air resistance was taken into account.

### Possible support

Work together as a class filling in the first few values of the time/distance table.