I went home for Christmas and as I was driving I got to thinking about speed and such. Hopefully y’all understand me.

Let’s say I was 60 miles out of a city and I’m driving 60 MPH. If I understand correctly that means I’ll get to the city in an hour. My question is, if at 59 miles out I drop down to 59 MPH, and at 58 miles out, I drop to 58 MPH etc. How long would it take me to get into the city?

Another question: A car is 10 miles ahead of me and we’re both going 60 MPH, how fast would I have to drive to catch up to him and how long would it take?

Thanks. These questions have been bothering me for a while, and I suck at math so it’d take me forever to figure it out.

Someone better is bound to come along with the formulas, but IIRC, in answering the first question, you’ll never get there. As for the second question, the formula is incomplete because you need to include at least one of the two remaining variables to come up with an adequate answer.

At 1 mile left you will be going 1 mile, and would drop to 0 mph when you arive - which is OK.

so you will be going 1 mile at 1 mpg, 1 at 2mph, 1 at 3mph…

The last mile will be 1 h
the 2nd to last would be 1h/2
the 3rd to last would be 1h/3
all the way to 1h/60

so 1+1/2+1/3+1/4+1/5… all the way to 1/60

I’m not going to work this out, but the 1st numbers will be the most significant.

Velocities are relative, make it easy you both are going 0 mph and are 10 miles apart.

To make up those 10 miles you will need to pick a speed:

Distance = speed * time
so
10h = speed * time
If you chose 10mph as your speed then your time will be 1 hour.

Now you just have to put this back into the orginal frame, so 10mph + 60 mph = 70mph (which is the speed you need to average for 1 hr to catch up). The distance you cover can also use that same equation
Distance = speed * time
Your distance will be your speed (70mph) * your time (1h) = 70 miles

Question 1: Since t=d/r, the first mile took 1/60 hour, the second mile took 1/59, etc. The total time is just 1/60 + 1/59 + 1/58 + … + 1 = 4.679870 hours.

Question 2: You’re asking for the answer to two unknowns – the rate and the time. No can do since the time will depend on how much faster you are driving. Let’s say you are driving k mph faster than the car ahead of you. Then you will catch up in 10/k hours. For example, if you’re driving at 65 mph, you will catch up in 10/5 = 2 hours.

It would be true that you would never get there if your drop in speed was done continuously, but in the problem as he states it you drop speed incrementally.

For the continuous version:

dx/dt=-x

where x is the distance to town.

OK, how do you do integral symbols? I will use |

|dx/x=-|dt

t=-ln(x)

You get to town when x is 0. That gives an infinite answer for t, the time it takes to get there.

Not homework. I wish I could be in school right now, but I can’t.

Thanks for all the replies. I had to read the thread a good ten time before I understood any of it (numbers are my enemy) but I finally did. Thank you all again.

Not sure how common sense leads you to 60 hours. That’s how long it would take if he drove 1 MPH the entire way. If he’s going any faster than 1 MPH at any point, it has to be less than 60 hours, doesn’t it?