Optimizing Race Kinematics: Solving for the Fastest Time

[student1ds]

Hi, I am unable to figure this question. Could someone help please ? A new type of race has been proposed. The object of the race is to complete the course and stop on the finish line in the minimum amount of time. The race is exactly 1 km long. The car can accelerate at 8m/s^2 and decelerate at 5m/s^2 when the brakes are applied. What is the best possible time to complete the race?

 

[student1ds]

I tried using the kinematics equations but the deceleration bit is throwing me off each time.

 

[qswdefrg]

hello person from Corsini's class.

... I don't get the question either. u__u;

 

[PhanthomJay]

Yes, its all in the kinematic equations and the fact that the car accelerates at 8 m/s/s over a distance d1, then decelerates over a distance d2 at 5 m/s/s, where d1 + d2 =1000 m. Hint: the car accelerates from 0 to some maximum speed, then decelerates from that same max speed to 0 at the finish line.

 

[student1ds]

I think I got an answer but to me it doesn't make sense. Is it possible to complete a 1 km race in 9 seconds ?

 

[qswdefrg]

Yes, its all in the kinematic equations and the fact that the car accelerates at 8 m/s/s over a distance d1, then decelerates over a distance d2 at 5 m/s/s, where d1 + d2 =1000 m. Hint: the car accelerates from 0 to some maximum speed, then decelerates from that same max speed to 0 at the finish line.

I think I'm on the right track... but I always seem to end up with more variables than equations. :/

 

[PhanthomJay]

I think I got an answer but to me it doesn't make sense. Is it possible to complete a 1 km race in 9 seconds ?

Not at an acceleration of 8m/s/s. Even if the car were to cross the finish line without applying the brakes before then, the time would be determined from d=1/2(a)(t)^2, solve t = almost 16 seconds...so since the car is braking well before the finish line in order to stop and have no velocity at the finish line, the total time must be greater than 16 seconds.

 

[PhanthomJay]

I think I'm on the right track... but I always seem to end up with more variables than equations. :/

One equation is d1 +d2 =1000; you'll then need a couple of the kinematic equations for each portion of the trip (d1 and d2, respectively), to solve for the time of each portion.