Race Car Scenario: Who will win?

[Zant]

Homework Statement

Our professor has tasked us to come up with a thought experiment to test the following scenario: "Two vehicles are going to race a fixed distance, both starting from rest. One vehicle has a greater maximum acceleration than the other, but a lower maximum speed. If both vehicles reach their top speed before completing the race, which vehicle will win?"

Homework Equations

We are allowed to use any of the following four equations (plus the gravitational constant):
Δx = 1/2(v₀ + v)t
Δx = v₀t + 1/2at²
v = v₀ + at
v² = v₀² + 2aΔx
g = 9.80 m/s²

Values for the first car:
Δx = fixed distance = 3300 m
v₀ = initial velocity = 0 m/s
v = final velocity = 240 km/h
a = acceleration = 3000 m/s²

Values for the second car:
Δx = fixed distance = 3300 m
v₀ = initial velocity = 0 m/s
v = final velocity = 280 km/h
a = acceleration = 2000 m/s²

The Attempt at a Solution

To be honest, these values were chosen by me. I have no idea how to determine which car would win in a 3.3 km race (3300 m) given the above values. What stumps me is which equation to use to determine the result of the experiment.

 

[TomHart]

If you are picking your own values, I would recommend that you use consistent units. So you may want to pick a final velocity in m/s. And since "both vehicles reach their top speed before completing the race", for each car, the first part of the race will involve some positive acceleration and the remainder of the race will involve 0 (zero) acceleration. So you have to deal with those two situations separately for each car.

 

[Zant]

So which equation would be best to use in this scenario?

 

[TomHart]

Which car wins? The one who completes the distance in the least amount of time. So time is something you need to find out for the whole race. However, once you get to the second portion of the race where the velocity is constant, you will have to know the distance of the final segment to calculate how much time it takes for that portion of the race. So once you know the distance and velocity (constant velocity) for the second portion of the race, you can calculate the time for that portion.

So for the first part, you have initial velocity, final velocity and acceleration. You need to find an equation that, given that information, you can solve for t and x for that portion. Those will both be used in the overall solution.

 

[SammyS]

Homework Statement

Our professor has tasked us to come up with a thought experiment to test the following scenario: "Two vehicles are going to race a fixed distance, both starting from rest. One vehicle has a greater maximum acceleration than the other, but a lower maximum speed. If both vehicles reach their top speed before completing the race, which vehicle will win?"

Homework Equations

We are allowed to use any of the following four equations (plus the gravitational constant):
Δx = 1/2(v₀ + v)t
Δx = v₀t + 1/2at²
v = v₀ + at
v² = v₀² + 2aΔx
g = 9.80 m/s²

Values for the first car:
Δx = fixed distance = 3300 m
v₀ = initial velocity = 0 m/s
v = final velocity = 240 km/h
a = acceleration = 3000 m/s²

Values for the second car:
Δx = fixed distance = 3300 m
v₀ = initial velocity = 0 m/s
v = final velocity = 280 km/h
a = acceleration = 2000 m/s²

The Attempt at a Solution

To be honest, these values were chosen by me. I have no idea how to determine which car would win in a 3.3 km race (3300 m) given the above values. What stumps me is which equation to use to determine the result of the experiment.

You will need to use more than one of those equations, perhaps all of them depending upon the details of your analysis.

Some of your values may not be realistic, particularly acceleration values. The fastest professional drag racing vehicles cannot achieve much above 1 g of acceleration (9.80 m/s²). They race for a distance of 1/4 mile, ≈ 0.4 km.
For those huge accelerations which you have chosen, the cars attain top speed in a few hundredths of a second or less.

So for a somewhat longer race you may want accelerations of 0.5 g (0.5 times acceleration due to gravity).

Tom Hart's suggestion is sound. 240 km/h ≈ 66 m/s.

You probably get the most helpful examples if the cars attain full speed somewhere in the middle 1/3 of the race.

 

[Zant]

Thanks! I'll mark this question as solved.

 

[billy_joule]

It's a thought experiment and the answer can be found without doing any math at all...
Choosing numerical values will only give an answer to a single set of variables, I don't think that's what your professor is looking for

 

[SammyS]

Thanks! I'll mark this question as solved.

It doesn't look solved to me.

 

[rcgldr]

The fastest professional drag racing vehicles cannot achieve much above 1 g of acceleration (9.80 m/s²). They race for a distance of 1/4 mile, ≈ 0.4 km.

Too fuel dragsters and funny cars pull over 4 g's. Due to the high speeds achieved (plus insufficient safe distance to brake), the distance was reduced from 1320 feet to 1000 feet. Wiki articles:

http://en.wikipedia.org/wiki/Top_Fuel

http://en.wikipedia.org/wiki/Funny_Car

These cars are close to top speed by 700 feet, since the last 300 feet take so little time (about .7 sec) when at close to 300 mph / 480 kph, and aerodynamic drag reduces acceleration to somewhere between 1 and 2 g's resulting in around 330 mph at the 1000 foot finish line.

 

[SammyS]

Too fuel dragsters and funny cars pull over 4 g's. Due to the high speeds achieved (plus insufficient safe distance to brake), the distance was reduced from 1320 feet to 1000 feet. Wiki articles:

http://en.wikipedia.org/wiki/Top_Fuel

http://en.wikipedia.org/wiki/Funny_Car

These cars are close to top speed by 700 feet, since the last 300 feet take so little time (about .7 sec) when at close to 300 mph / 480 kph, and aerodynamic drag reduces acceleration to somewhere between 1 and 2 g's resulting in around 330 mph at the 1000 foot finish line.

I stand corrected.

I think that you will agree that the accelerations given in the OP are way too high: something like 200 and 300 g's . Not survivable by a living driver.

 

[billy_joule]

Our professor has tasked us to come up with a thought experiment to test the following scenario: "Two vehicles are going to race a fixed distance, both starting from rest. One vehicle has a greater maximum acceleration than the other, but a lower maximum speed. If both vehicles reach their top speed before completing the race, which vehicle will win?"

Lets take a step back and look at the two simpler cases
1) where a₁ =a₂
obviously the car with the greater top speed will win, regardless of d_{race length}

and 2)
where v₁ =v₂ (where v is the max velocity)
obviously the car with the greater acceleration will win, also regardless of d_{race length}

Now, if try to answer your professors (purposefully vague, I suspect) question, we have 5 variables; a₁, a₂, v₁, v₂ and d_{race length}
And all we know is that:
a₁ > a₂ and v₁ < v₂

Can we find a solution where t₁ > t₂? Or t₁ < t₂? Or t₁ = t₂?
Or all three? How many of each can we find?

 

[rcgldr]

There's not enough information to provide an answer.

Assuming constant acceleration and that the faster accelerating car top speed is at least greater than 1/2 of the top speed of the slower accelerating car, then at the distance that the slower accelerating car reaches its top speed, the faster accelerating car would be first, having already have crossed that distance. There would be some distance that both cars reach at the same time, and beyond that distance, the slower accelerating car with higher top speed would be first.

However, the problem only states that one car has greater maximum acceleration. This could be due to having stickier tires allowing lower gearing on a car to provide faster initial acceleration, even if the car with stickier tires has a lower power to weight ratio, resulting in a lower average acceleration than the other car.