Sitting duck

Problem 191 — It sometimes pays to be at right angles with your relations, as this month’s problem by Jannine Sobottka of Mesilla Park, N.M., demonstrates.

“Why, what wonderful weather for a day in the park,” chirped Phineas Gotrocks to his 13-year-old nephew. “Here, Spaulding, make a video of me riding the Ferris wheel!”

Gotrocks climbed into his seat, and the ride started. He chortled as he noticed his good friend Lottie Dough with a gentleman not her husband on another seat.

Meanwhile Spaulding had quietly put down the camera and picked up a large slingshot and water balloon. He prepared to fire the balloon. The Ferris wheel has a 20-ft radius and turns at 1 rpm. Lottie and her companion had rounded the top and were descending, as Spaulding fired his balloon. Gotrocks was nearing the bottom of the ride.

Spaulding stood with his slingshot 75 ft from the point directly beneath the center of the wheel on the side where the riders descended. The balloon’s initial velocity was 60 fps, and it had been fired at an angle of 60 deg. The slingshot was at the same elevation as the bottom of the wheel seat.

Neglect wind resistance and the fact that a standard water balloon would probably break at 60 fps. Who gets wet: Lottie and her boyfriend, or Gotrocks? Or does the balloon fly up and over the Ferris wheel without hitting anyone?

Solution to last month’s problem 190 — You have your affairs in order if you answered 312 ft. Here’s how to get a lift:

Let:
V = Volume of balloon, given as 10,997 ft3
Wfw = Weight of Wurme, given as 150 lb
Wb = Weight of balloon and basket, given as 500 lb
CD = Drag coefficient, given as 0.20
v = Velocity of balloon, fps
F = Net lifting force of balloon, lb
d = Diameter of balloon, ft
Phyd = Density of hydrogen, given as 0.006 lb/ft3
Pair = Density of air, given as 0.0802 lb/ft3

You can express the aerodynamic drag force on an object in terms of air density and the object’s size and velocity:

For a gas-filled balloon, the net aerodynamic force is its lift force:

Calculate the balloon’s diameter, d, from its volume:

This is the balloon’s terminal velocity, which is reached after about 2 sec. Thus, a valid approximation is 5.2 fps × 60 sec = 312 ft. Wurme climbs 312 ft in one minute and comes down with a bang!

Contest winner — Congratulations to Paul Ornosky of Proctorville, OH, who won our November contest by having his name drawn from the 73 contestants who answered correctly out of a total of 105 entrants for that month. A TI-68 calculator is in the mail to him.

The TI-68 Advanced Scientific Calculator by Texas Instruments can solve five simultaneous equations with real and complex coefficients and has 40 number functions that can be used in both the rectangular and polar coordinate systems. Other functions include formula programming, integration, and polynomial root finding. The calculator also features a last-equation replay function that lets you double-check your work.