Math can Launch One Bus with Speed minus Gravity


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Speed and Distance in Freefall : Image by Mike DeHaan

Could a bus really cross a chasm? Although the Mythbusters have already tested this aspect of Jan de Bont’s movie, Speed, in real life, let’s use math to drive through the calculations.

The Physical Constraints to Launch a Flying Bus

Two important physical constraints were imposed by the movie. In Speed, the bus had to cross a fifty-foot-long gap in an overhead highway ramp, and it had to maintain a driving speed of at least 50 miles-per-hour.

The real world imposes another constraint; that the downward acceleration of gravity is 9.8 meters, per second squared. Gravity’s acceleration does not depend on the mass of the object, so we can ignore the weight of the bus, fuel, bomb, and passengers.

Although a typical bus is about 40 feet in length, let’s simplify this condition in the next paragraph.

Simplify the Bus by Rounding it into a Cannon Ball

It will be easier to estimate the chances of the bus making it across the chasm by visualizing it as a ball. Image by ozz13x

A real, physical bus presents problems for this mathematical model.

In the real world, the front of the bus would start to rotate down as soon as the front wheels left the horizontal road. In the movie, however, the nose of the bus rotated sharply upwards as the front bounced off the small ramp built for this effect shot. In reality, this would be difficult to calculate and predict. It would be especially bad if either the undercarriage or the rear bumper were to scrape against the road as it went off the edge.

A bus is not very aerodynamic, so it would decelerate once airborne, and so it would spend longer accelerating downward. It’s hard to imagine that the bus would have any “lift” from its shape to counteract the fall, but possibly the nose-up bus did experience lift due to the ramp.

Instead of calculating for the oddly shaped, ungainly bus, let’s simplify the calculations by treating the bus as a cannon ball. That allows us to ignore aerodynamics and nose-first dives, and to deal with only single moments for takeoff and landing.

Convert All Units to Feet and Seconds

Let’s do all the arithmetic in units of feet and seconds.

  • At 39.37 inches to the metre, gravity’s acceleration of 9.8 metres per seccond-squared equals 32.15 feet/second^2.
  • There are 5280 feet/mile, and 60*60=3600 seconds per hour. Therefore to convert from miles-per-hour to feet-per-second, we multiply by 5280/3600.

The Time Constraints to Launch Across the Gap

At the end of the “Speed” video on YouTube, the speedometer reads about 56mph after landing. Here’s the video:

Let’s assume that our simplified cannon ball launched at 60mph and retained that speed.

  • We convert 60mph * 5280/3600 =  88 feet/second.
  • The cannon ball would cover 50 feet in 50/88 = 0.568 seconds, assuming a flat horizontal trajectory (as if rolling).

In this scenario, the minimum time to cross the gap in the highway is 0.568 seconds.

The Worst Scenario: No Ramp and a Level Gap

The formula for the distance fallen solely due to the acceleration of gravity over a period of time is “distance = (1/2)*gravity*(time^2)”. Without a ramp, the bus would have no initial upward motion, and would fall as it moved forward. In 0.568 seconds, gravity would take the bus down 5.18 feet.

(1/2)*32.15*0.568*0.568 = 5.18 feet.

With a level gap, and without a takeoff ramp, the windshield of the bus, rather than the wheels, would meet the pavement. Ouch. Our bowling ball would simply miss the highway altogether and plummet to its doom.

The First Safe Solution is a Lower Runway for Landing

The most obvious safe solution is to build the next section of highway at least 5.2 feet lower than the upper. The landing would be painful, but at least the bus would be on top of the road surface, rather than impaled through the windshield. The downward speed of the bus would be 18.26 feet per second.

32.15*0.568 = 18.26 feet/second.

Imagine riding a bus that dropped more than five feet as you were driving!

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