Runner
It’s a sunny Sunday afternoon and you are walking around a large lake. A runner approaches you wearing a T-shirt that you try to read as he runs past. You didn’t finish reading the T-shirt, so you wonder, if the runner continues around the lake, when you will see him again.
You look at your watch and it is 3:07 p.m. You recall the lake is 3.3 miles in circumference.
Is it likely you’ll see the runner again?
Hint
Reveal

Hint
This is not a complicated problem, but you’ll need to make some estimates here. How fast does someone walk / run?Solution
Reveal

Solution
Let’s estimate your average walking speed at 3 miles per hour and the runner’s average speed to be about 7 miles per hour.
It’s quite likely you will see each other again. In 20 min you will have walked 1.0 mile and the runner travels 2.3 miles.
There are at least three ways to solve this problem (this is the easy way - almost by inspection and by choosing ’nice values’).
You can also (2) set up the algebraic equations and solve for the unknows of time and distance, or (3) treat this as a relative motion problem, recognising that the situation is the same as the walker standing still and the runner moving at 10mph.
In all cases, the key thing is to work through the problem solving strategy (FDPEE!)
- Focus on the problem: draw a sketch, identify unknowns and define them. Do we have all the information needed (no, so estimate it)
- Describe the physics: straighforward here - it’s just distance = speed x time
- Plan a strategy to solve it and set up the equations
- execute the strategy: work out the answer
- evaluate the answer; does it seem reasonable (easy to do this, since we all ’know’ about walking and running!)
To work through method (2) .....
- .. starting at the plan a strategy step of FDPEE....
- Let d,v be the walker’s distance covered, speed of walking respectively
- Let D,V be the runner’s distance covered, speed of walking, respectively.
- Let t be the time (common to both!) before they both meet
- Estimate v=3mph and V=7mph and we know d+D=3.3mi
- Strategy is to equate the expressions for the the time taken to meet for both runner and walker and solve
- Setting up the equations:
- Solving: 10-3d=7d⇒10=10d⇒d=1mi
- So they meet up after the walker has travelled 1 mile, which takes 20 minutes.