## Sunday, June 20, 2010

### Happy Father’s Day

Happy Father’s Day to all the dads out there.  My Dad suggested a problem the other day: How long would it take to wear down a tire by 0.75 inches if you’re driving a 55 mph?

I’ll assume a tire tread starts about an inch thick.  By the time 0.75 inches have worn off, it’s time to get them replaced.  Tires presumably wear down because of friction between the rubber and the road.  When tires rotate, this frictional force from the ground pushes the car forward while an equal and opposite force pushes back on the road.  The faster you drive, the greater the frictional force is and this presumably causes more wear and tear.  However, if you’re driving at a constant speed, you probably don’t burn as much rubber as you would by slamming on the breaks every five seconds.  It’s not immediately clear which of these effects causes greater erosion, but you’re generally advised to change tires roughly every 50,000 miles1.  If that’s the case for driving at 55 mph, then it’s easy to compute how long it will take before your tires have worn down by 0.75 inches:

time = (distance) / (speed)
= (50,000 miles) / (55 mph)
= 38 days.

It should take a little more than a month to wear off 0.75 inches of your tires by driving at 55 mph.