In: Math
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?
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.]