Tuesday, June 12, 2012

You Do The Math - Driving Golf Balls On The Moon

You Do The Math - Driving Golf Balls On The Moon

Where can you drive a golf ball farther, on the Earth or the Moon?  I lost a point in a trivia contest because I had once read that you can drive a golf ball farther on the Earth, due to the fact that backspin provides extra lift.  Turns out, this is not so!  And, thankfully, since there is no air on the Moon, a little physics 101 works just fine,  So today...  You Do The Math!

Here are the results of what we will eventually calculate.  Tiger Woods' average drive is just over 300 yards, or 900 feet.  This drive is in pink.  Tiger has a 125 mph swing, giving the ball a speed of 185 mph.  His 8.5 degree driver gives the ball an 11 degree initial trajectory.  As you can see, the Earth's atmosphere let's it continue to rise, but also robs speed.  This is similar to how an airplane flies when slowing down for a landing.  As you go slower and slower, you need to pull back farther on the stick to pitch the plane higher thus maintaining enough lift to stay in the air.  And...   most importantly, as you pull back on the stick, you create more drag, slowing the plane even more.  Eventually, the speed is too slow to provide enough lift to fight gravity, and you begin to descend.  This descent accelerates so rapidly the trajectory of the plane or ball tries to approach vertical, like in the pink curve.

On the Moon there is no air, so the ball will be a simple projectile, hitting the ground at the same angle and same speed it had when it left the tee, no matter what kind of spin it has.  With an 11 degree initial angle, and a speed of 185 miles per hour, the ball will go over 5000 feet, and that's a LOT more than Tiger's average drive of 900 feet.  In fact, it's a couple hundred feet shy of a mile!   And what about the bounce?  The Moon's surface is powder, and a ball should dig in pretty well.  But at 185 mph and a shallow angle like 11 degrees, I'm thinking it will skip a long way.  It might not skip over a lunar lander like the Eagle, but might skip past the "green".  

By "green" I mean sand trap.  In fact, the entire course will be a sand trap.  And that would be one HUGE golf course, many miles long for each hole, as we shall see below.  But, those drives will sure make you feel proud.

In physics we learned that a projectile will cover the most distance when fired at a 45 degree angle.  This is because the sine and cosine of 45 are equal, and are both 0.707.  This means the ball has a vertical speed of about 71 percent of that 185 mph.  It also has a horizontal speed of 71 percent of 185 mph.  Many times in science the best results occur when input factors are not too great and not too small, but right in the sweet zone, like the one on a baseball bat.  This is certainly true here.  

When fired at 11 degrees, the horizontal speed is 98 percent of 185, while the vertical speed is only 19 percent of 185 mph.  Hang time is very small.  Another way of looking at it, we only get 98 percent added to 19 percent, for a total of 117 percent of the initial speed put to good use.  However, 70.7 percent plus 70.7 percent is 141.4 percent.  Looks like we are using more energy than we started with.  (This type of weird thing also happens in physics a lot.)

To figure out the distance of the drive, you multiply horizontal speed by hang time.  To figure out hang time, fist get vertical speed by multiplying the initial speed by the sine of the angle.  In a "right triangle", the sine of an angle equals the opposite over the hypotenuse.  In other words, the length of the side of the triangle opposite the angle in question, divided by the length of the hypotenuse.  The hypotenuse is the side of the triangle that is NOT touching the right angle.  (The right angle is the 90 degree angle.)   

Once you have vertical speed, divide it by the Moon's acceleration due to gravity.  This is approximately 1/6th of Earth gravity, which varies significantly depending on your latitude.   Because the Earth's rotation gives everything on the equator a speed of 1000 mph, the Earth bulges a little in the mid section.  By a "little" I mean about 6.6 miles.  This means you are 6.6 miles farther from the center of mass of the Earth, so gravity is a bit less than on the poles.   The centrifugal force subtracts a bit, as well.

Whatever, lets use 32.174 feet per second squared.  This means that for each second of flight the ball looses a speed of 32.174 feet per second.  Take 1/6th of that for Moon drives, and we get 5.36 feet per second lost per second of time.

To convert Tiger's 185 miles per hour to feet per second, you multiply by 5280 feet per mile, then divide by 3600 seconds per hour.  In other words, at 1 mph you go 5280 feet in 3600 seconds.  This is equal to 88 feet in 60 seconds, or about 90/60, or 3 over 2, or 1.5, if you want to do this math in your head.  (So, for you motorheads, if you are driving 60 mph, you are going 88 feet per second.  The legal assured clear distance is 2 or 3 seconds, about the combined length of 4 and 1/2 semi tractor trailers.  Much more than 2 car lengths.)

Here's an epiphany due to a mistake I just made doing the math.  What if the Earth had no air?  How far would Tiger drive the ball?  Vertical speed for the driver is the initial velocity of 185 mph times the sine of 11 degrees, which comes to 35.3 mph.  This is equal to 51.8 feet per second.  For the Earth, divide by 32.174 feet per second lost for every second of flight, we get a time of 1.609 second to maximum altitude.  Hang time is twice that, since there is no air, so the ball "flies" for 3.22 seconds.  (Actually, no air means no flight, it's a simple projectile, hence the use of physics 101.)

Now we multiply 3.22 seconds by the horizontal speed, which is the cosine of 11 degrees times 185 mph, which is 0.98 times 185 times 3.22, or 181.6 mph times 3.22 seconds, or 266.35 feet per second times 3.22 seconds, or 857.2 feet.  That's less than Tiger's 300 yard drive.  No wonder people think you can drive it farther on the Earth than the Moon.  Another conundrum solved!  (See my blog on Gabriel's Horn, the first conundrum I solved.)

Now let's do the calculations the right way, with Moon gravity and no air.  Hang time is 35.3 mph divided by a measly 5.36 feet per second lost per second of flight.  This is 9.65 seconds, so hang time is twice that, 19.3 seconds.  That's 6 times more than on an airless Earth.  I wonder what will happen?  You guessed it.  Tiger drives it 6 times farther on the Moon, a whopping 5143 feet.

However, as we stated before, without air, the farthest you can throw or launch anything is using a 45 degree angle.  What club does that?  In the diagram below, we see that the 5 iron comes close to that, assuming there's a constant relationship between head angle and the ball's angle of flight, 7.36 percent.  But, that's a big assumption.  Instead of assuming the 5 iron is the right choice, we can simply use all the irons in the bag to figure the maximum and minimum speed of a drive with an iron.

And another big assumption is that the same increase in velocity occurs as would a drive with Tiger's driver, from 125 mph to 185 mph, or 48 percent.  Using only the second assumption, applying it to the longest and shortest irons in the bag we will get a maximum and minimum drive speed.  With these speeds, and using a 45 degree angle, we will get a range for the maximum possible drive on the Moon.

Measuring the length of my swing (crudely) shows that my driver swing is 5 feet 11 inches, my longest iron swing is 5'4" and my shortest iron swing is 5'1".  So, we are comparing swings of 71 inches, 64 inches and 61 inches.  This ratio is multiplied by the 125 mph swing, reducing it.  Then we assume a 48 percent increase for the ball's speed.  Convert to feet per second, then use a 45 degree trajectory, divide by the acceleration due to gravity, double it, and we get...

Hang time of shortest iron:
125 mph * (61/71) * (185/125) * cosine (45) * (88/60) * (6/32.174) * 2 = 61.48 seconds

Hang time of longest iron:
125 mph * (64/71) * (185/125) * cosine (45) * (88/60) * (6/32.174) * 2 = 64.5 seconds

Drive length of shortest iron:
125 mph * (61/71) * (185/125) * sin (45) * (88/60) * 61.48 seconds = 10,134 feet

Drive length of shortest iron:
125 mph * (64/71) * (185/125) * sin (45) * (88/60) * 64.5 seconds = 11,155 feet

That's two miles plus or minus 10 percent.  Remember what I said about BIG golf courses on the Moon?  Even a par 3 hole on the Moon would be 2 miles long.  A par 5 hole with 2 good drives would be 4 miles long.  A 9 hole golf course, averaging 3 miles each, would force you to walk (or skip, like an astronaut) a total of 27 miles.  Good thing hopping is easy in 1/6th gravity.

The only problem...   with a pressurized and bulky space suit, drives are one-handed, and only go 200 to 400 yards, according to Al Shepard.  His video can be found on the internet.

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