From the Book
“So according to this,” Will said, flicking back to the skid calculations. “Jeff was driving at eighty kilometres an hour, braked hard for thirty metres and still hit the wall at seventy-seven kilometres an hour.”
He squinted at the page and then jabbed the formula. His heart thudded in his ears. “Except he didn’t. They’ve done a NASA.”
“What?” Drake said, frowning and peering at the formula.
“Mixed up the units,” Eliza said. Her voice rose as she leaned forwards to the report. “NASA mixed up miles and kilometres. Whoever did this – ”
Will and Eliza are at the police station going through the investigation report for the crash in which Will’s uncle Jeff died. They’re examining the calculations for the car speed and its impact velocity with the tunnel wall.
Determining the speed of Jeff’s car
The crash report has surveillance camera footage of Jeff’s car moments before the crash. The two camera frames are from one second apart, and by measuring the road the police have estimated the car has travelled about 22 metres over that time period.
Car speed = distance travelled / time taken
Car speed = 22 metres / 1 second
Car speed = 22 m/s
To get the speed in km/hr, we need to convert from metres to kilometres by dividing by 1000, and from seconds to hours by multiplying by 3600:
Car speed = 22 m/s x 0.001 m / km x 3600 s / hr
Car speed = 79.2 km/hr
Determining the impact speed
To calculate the impact speed of the car with the wall, the report uses the formula:
- v_initial is the initial speed of the car
- g is gravity
- d_braking is the estimated braking distance, which can be estimated from the length of the skid marks – about 30 metres.
- f is the friction coefficient. The report has used a coefficient of 0.7.
- G is the road gradient (the slope of the road) – in this case 0.1, since the road is slightly uphill
The report plugs the numbers into the formula like this:
Eliza realizes something’s up – braking for 30 metres and only reducing the speed by less than 3 km/hr doesn’t make a lot of sense. After a bit of thinking, they realize that the report has mixed up different units in the formula:
- The initial speed of the car has been entered in units of km/hr
- The values for gravity and the skid mark length have been entered using units of metres (m/s/s for gravity, just m for the skid mark length)
Eliza re-calculates using consistent units – metres:
With consistent units, they calculate that the car had slowed down a lot by the time it hit the wall. This new information adds to their suspicion about the report.
Real Life Example – Speed Cameras on Downhill Roads
A lot of people complain when police put speed cameras on downhill roads. Let’s see what happens if we add a downhill gradient of 0.1 to the calculation above:
When the slope is downhill, it takes longer to slow down – after a 30 metre skid the car has only slowed to 41 km/hr (instead of the 13 km/hr in the uphill case).
A 28 km/hr speed change could mean the difference between a minor injury to a pedestrian and a major injury or even fatal accident.
So speeding downhill can be even more dangerous – because if you have to slam on the brakes, it can take longer to slow down to a safe speed or to stop.