Some engineers I know analyzed the Lexus commercial (the one that shows a car falling from the sky and a Lexus on the road passing under it just before the falling car hits the ground). The commercial says something like “Gravity will propel this (falling) car 4000 ft in a matter of seconds. The Lexus (on the road) will cover the same distance in less time.” There is a caption (in very small print) at the bottom of the screen during the commercial that reads “Based on a horizontal drop.” In other words, this maximizes drag and increases the time it takes for the falling car to hit the ground.
Here’s the commercial on you tube if you haven’t seen it.
http://www.youtube.com/watch?v=gsZ87yeJw8M
For the Lexus on the road, they used real data from the Lexus website. Here were the 3 pieces of info we needed.
1.Goes from 0 – 60 mph in 5.3 sec
2.Covers ¼ mi (starting from rest) in 14.2 sec
3.Top Speed: 140 mph
Based on this info., we calculated the following:
The Lexus hits it top speed of 140 mph in 16 sec. It travels 1670 feet during this time. The remaining 2330 ft (4000 – 1670 = 2330) takes 11.4 seconds (with the Lexus going at a constant 140 mph)
This means that the total travel time for the Lexus is 27.4 seconds (16 + 11.4 = 27.4)
Now, for the car falling from the sky. This is more tricky because you have to consider drag and also terminal velocity. We calculated that the terminal velocity of the falling car is 117 mph. This value takes into account drag.
Now, using that value but neglecting drag when using the kinematic equations, we calculated that it takes the falling car 5.3 seconds to reach its terminal velocity. The car only travels 452 ft during this time. During the remaining 3548 feet of the fall (4000 – 452 = 3548), the car is traveling at a constant velocity of 117 mph. It takes the falling car 20.8 seconds to cover this distance.
Thus, the total falling time is 26.1 seconds (5.3 + 20.8 = 26.1)
Looking at the two bold lines above, we can see 26.1 s is just slight less than 27.4 s. In other words, the car would hit the ground 1.3 seconds before the Lexus on the road gets to that point.
But, as mentioned above, we neglected drag during a portion of our calculation.
The drag calculation depends on the square of the velocity. But since the velocity of the falling car is rapidly changing during the initial portion of the fall, a true calculation of the drag force would require calculus. Unfortunately, it would have taken to much effort, so we decided to just use an average velocity instead of going through the differential equations.
The revised calculation (including drag) yields the following:
The falling car takes 7.3 seconds to reach its terminal velocity (instead of 5.3 sec). It covers 623 feet during this period (instead of 452 ft). The inclusion of drag during this portion of the fall yields an acceleration of 23.4 ft/s^2 (instead of the conventional gravitational accel. of 32.2, which is what we used before).
The remainder of the fall takes 19.8 sec (instead of 20.8 sec).
And so, the total falling time is now 27.1 seconds instead of 26.1 s.
Therefore,
Total Falling Time: 27.1 seconds
Total Driving Time for the Lexus on the Road: 27.4 seconds
And so, the 2 cars travel 4000 feet in about the same amount of time, and the commercial is accurate.
The biggest factor here is the previously mentioned caption that reads “Based on a horizontal drop”. The horizontal drop yields a huge drag coefficient, which results in a very large drag force. This huge drag force is why the falling car hits a terminal velocity of only 117 mph.
What the commercial doesn’t say is that it is impossible for the car to remain in a completely horizontal position during its entire fall. Just for the sake of comparison …. if the call were falling vertically instead of horizontally, the terminal velocity would be 313 mph (instead of 117 mph), and it would take a distance of 4500 feet to reach this velocity (remember, the car is being dropped from only 4000 ft).
It would take the car 18.5 seconds to hit the ground (instead of 27.1 sec).
In other words, the car would hit the ground 9 seconds before the Lexus on the road would hit the same point. The bad news (for Lexus at least), would be that the commercial would be completely inaccurate. The good news is that the guy driving the Lexus would have plenty of time to avoid a collision.
I think to correctly analyze this you must factor in "buyer-beware marketing" and "free market capitolism".