You are the world’s best driver and pilot for setting world speed records. Car, plane or boat – you’re the one who sets all the records. High end companies fly you all over the world to set records in their new equipment, and pay you big dollars if you succeed. But behind the scenes, it takes a lot of preparation and calculation to make sure that you can perform the record attempt safely, and live to race another day.
Math Scenario 1: Quarter mile in Singapore
Your first gig for the year is in Singapore, where you’ve been commissioned to test a new supercar model that will come out later in the year.
The car needs to do the 280 m stretch in under 9 seconds for publicity purposes. It doesn’t have a constant acceleration though – it’s faster to accelerate from 0 to 100 km/hr than from 100 km/hr onwards.
To simplify matters, let’s assume that the car has a constant acceleration from 0 – 100 km/hr, then has a constant (but slower) acceleration for the 100 km/hr – 240 km/hr segment.
This graph shows what the speed profile of the car would look like over time (a more sophisticated analysis would produce a graph that slowly curved over time rather than having the sudden reduction in acceleration at 100 km/hr):
The area under this graph represents the distance travelled.
Can the car make it 280 metres in under 9 seconds?
We can first work out how far it will travel accelerating from nothing to 100 km/hr:
d = 0.5 * a * t^2
Work out a:
a = 100 km/hr * 1000 m/km / 3600 s/hr / 2.5 s
a = 11.11 m/s/s
Back to the original equation:
d = 0.5 * 11.11 * 2.5^2
d = 34.72 m
That leaves over 245 metres to go. We can write an expression for the time taken to travel the remaining distance:
d = v_initial * t + 0.5 * a * t^2
This is a quadratic equation: we can rearrange into the standard form and solve for t:
0.5 * a * t^2 + v_initial * t – d = 0
Solve using the quadratic equation. First we need to know a:
Back to the quadratic. I’m going to use an online quadratic equation solver this time from math.com. Don’t get confused by the a for acceleration, which is different to the coefficient “A” in the standard quadratic equation form:
Ax^2 + Bx + C = 0
We have A = 0.5 * 5.401 = 2.701, B = 100 km/hr = 27.78 and C = -(280 – 34.72) = -245.28
Using the online solver, we get:
t = 5.686, -15.97 seconds
We can discard the negative example, which leaves a time of 5.686 seconds for the second section of the race.
Total time = 0 – 100 km/hr time + completion time
Total time = 2.5 + 5.686
Total time = 8.186 seconds
Well under the 9 second barrier!
Math Scenario 2: Fighter jet test pilot
A defence contractor has just completed the prototype of a new fighter jet they’re hoping to sell to some large countries.
For their demos, they need to show off the new jet’s speed in a straight line. Observers will watch from a passenger jet flying alongside. Unfortunately the passenger jet is much slower (it can’t break the sound barrier), so observers will only have a very limited time to observe your jet.
The observers will need about a 30 seconds to get up from their seats and get to the viewing windows of the passenger line.
If the observer plane is flying at 800 km/hr and your jet is flying at a top speed of 1900 km/hr, both on the same course, when should the observers be alerted to take their viewing positions?
To work this out, we can first calculate the effective closing speed – the difference between the fighter jet and passenger jet velocities. We can then work out what distance would be closed during the 30 seconds:
You’ll need the observers to start assuming their positions when your fighter jet is still more than 9 km away! That’s how fast these machines can fly (and remember the observers are in a jet themselves).
Real Life Example: Top Gear Bugatti Super Sport Speed Test
One of the most famous episodes of the long running British TV series Top Gear was when they speed tested the Bugatti Super Sport, one of the fastest production cars in the world. You can view that part of the episode below:
One of the challenges in maxxing out a supercar’s top speed is that you need a very long track in order to get to the very top speed, since the car’s acceleration gets quite low when near top speed.
Let’s say the top speed of the car is around 430 km/hr, but that it takes a full 30 seconds to get from 400 km/hr to 430 km/hr. How far will the car travel during this time?
If we assume that the car accelerates at a constant rate from 400 to 430 km/hr, then we can assume that its average speed over the time period if halfway between the initial and final speed – 415 km/hr.
Then it’s easy to work out the distance travelled:
d = average speed * time
d = 415 km/hr * 30 seconds
d = 415 km/hr * 30 seconds * 1 hr / 3600 seconds
d = 3.458km
That means you’d require three and a half kilometres of track just to do that final bit of the acceleration, not to mention all the track required to get up to 400 km/hr in the first place.
You can view an example of a top secret racing track using Google Maps below – which shows the Volkswagen Group test track at Ehra-Lessien in Germany: