# Motion of a sports car

1. Mar 2, 2015

1. The problem statement, all variables and given/known data

A sportscar, Electro-Fiasco I, can accelerate uniformly to 100 km/h in 3.5 s. Its maximum braking rate cannot exceed 0.7g. What is the minimum time required to go 1.0 km, assuming it begins and ends at rest?

2. Relevant equations

$$\Delta x = \int_{t_1}^{t_2} v(t) dt$$

3. The attempt at a solution

The problem with this question is that it describes what can happen, without describing what actually does happen. I know I could just plot velocity as a function of time such that the curve looks like a triangle with height $100 km/h$ and just solve for its base, $T_{min}$, by equating its area to $1000 m$, but I can't really figure out why I have to do that; it feels like guesswork. Even if I assume it accelerates uniformly in the beginning, how do I know when it stops accelerating? How do I even know that the acceleration is constant throughout the first part of the journey?

2. Mar 2, 2015

### PeroK

Ask yourself this. If the car accelerates and decelerates at its maximum rate (from and to rest), can you show that (in the same time), it cannot possibly go further? Hint: look at your triangular speed/time graph.

3. Mar 2, 2015

### jbriggs444

MohamedRady97, as I read your original post, you are already understand that the best approach is to accelerate as hard as possible as long as possible and then, as the last moment start braking as hard as possible so that the car comes to a stop exactly at the finish line. But you are unclear on exactly how to know when (or where) the braking has to start.

There are at least two ways to proceed. Since you do not know the time when braking has to begin, try making that time a variable, $T_{brake}$. Given what you know about the problem, what equations can you write involving $T_{brake}$?

[Alternately, you could try finding the position where you have to begin braking using $x_{brake}$ as the unknown value]

4. Aug 26, 2015

### AliGh

I was reading "An introduction to mechanics" last night and i ran into this problem (page 44)
First you have to prove that the best way would be that if you use all engine and brake power available
I don't have much knowledge of calculus or mechanics so sorry if something doesn't make sense
Suppose that the time spent accelerating is T1 and the whole time is T
The average V from time 0 to T1 would be 1/2*a*t . The point you are starting to brake your speed would be a*t and at the end of braking it would be 0 so the average speed on the time time of braking would be 1/2*a*t so the average speed for the whole time would be 1/2*a*t . Now a*t is the maximum speed the higher the maximum speed the higher average speed the lower time ... And what is the best way to reach the highest possible maximum speed ? Use all acceleration power you have as long as you can and in order to make it the longest period of time possible you need to use all your brake power