Question

I am stuck in Austin with a flat tire, and I need to get to my class in Houston within 2 hours. The drive is 180 miles long, but I want to be careful of the Austin cops. I do not usually get caught speeding unless I am seen accelerating too fast, so I do not want to accelerate at a rate of more than 120 mi?/?h². Just to be careful, I?m going to take exactly 2 hours to make the trip. Assuming I accelerate at 120 mi?/?h² for a while, and travel at a constant speed afterwards, what’s the fastest speed I?ll be going during my trip?

Answer 1

Since I am starting from Austin, it would mean that initial speed would be 0.

The acceleration is 120 miles/h² . Now assuming that I accelerate for t hrs and then travel up to Houston at a constant speed, the distance travelled and time taken to cover both parts of journey, can be calculated as follows:

In time t hrs I will attain a velocity given by the 1st equation of motion v= u +ft. Since u=0, f= 120miles/h²   , velocity at the end of time t would be 120t miles per hour. Now distance travelled in time t would be given by the second equation of motion S= ut + 1/2 f t² = 1/2 (120)t² = 60t² miles

The situation now is that I have travelled a distance of 60t² miles while accelerating. The remaining distance to Houston is 180-60t² miles, which I would cover with a constant speed of 120t miles per hour.

The time to cover the distance 180-60t² miles at a constant speed would be distance /speed =(180-60t²)/(120t).

The total time of the journey would be t+ (180-60t²)/(120 t) = 2hrs. Solving this eq for t would get us the time for which I would be accelerating. Multiply both sides by 120t, the eq would be 120t² +180-60t² = 240t

Or, 60t² -240t +180=0

Or, t2 -4t +3=0 -> (t-1)(t-3)=0. So, t= 1 or 3 hrs. Now reject the value t=3, because total journey is 2 hrs only. S0 t=1 hr.

The fastest speed would be the speed, that was attained after time t. At the end of time t, that was 120 t miles perhour or 120 miles per hour.

[The distance travelled while accelerating would be 60 miles and the distance travelled at constant speed would be 120 miles. Both parts of jouney would be of 1 hr each.]