How fast would I have to drive to keep up with an aircraft doing 100MPH and 500 feet above the ground?

Question

I would appreciate the formula if there is one

Answer 1

Well, the Earth is round, but adding 500 feet to an object 4000 miles in radius isn't much of a arc. You have to be moving 100 mph as well.

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I understand you're asking this because of the curvature of the Earth - most of the posters here are missing that. So, let's figure that part out:

If the plane is 500 feet up, that's about 0.1 miles. Therefore, the radius the plane is following is .1/4000, or 1/40000 larger than the radius you are following. You "only" need to go 39999/40000 as fast as the plane, which is... um... still about 100mph. (For the anal retentive, it's 99.9975mph.) By the way, I see a couple of posters have since contributed to this question with good analyses, except I think they are working too hard by calculating the respective circumferences of the two circles. This is unnecessary because the radius and circumference increase by the same ratio; in other words, the 2π's cancel out.

Now, if instead you are trying to stay under something moving 100mph that is 1000 miles up (a satellite, let's say), then the radius change is significant enough to make a difference. It become 4000/5000, or .8, so you only have to go 80mph.

Answer 2

First, we need to make several assumptions and accept them as given: the car is traveling on a continuous road – there are no interruptions of any kind and the car and driver will not need to stop for any reason. The airplane will always maintain a constant elevation of 500 feet above the car. There are no mountains, lakes rivers, oceans or other obstructions to interfere with either vehicle.
Consider two circles. The first is the circumference of the Earth. Using a diameter of 7,900 miles. (it's actually a little larger than this, but let's use a simpler number!), the distance the car would travel around the surface of the Earth on a continuous road is 7,900 Miles X 3.1416 = 24,818.58 Miles..
The second circle, concentric to the first circle and having a common center point, is the flight path of the airplane 500 feet above the first circle, so its diameter is 1,000 feet larger. Converting to feet, 7,900 X 5,280 = 41,712,000 + 1000 = 41,713,000 feet / 5,280 feet per mile = 7900.189 miles Diameter of the Earth X 3.1416 = 24819.18 Miles.
If the airplane maintains a constant speed of 100 Miler per Hour, it will take the airplane 24819.18 Miles / 100 Miles per Hour = 248.1918 Hours to travel the circumference of the Earth.
The car will then need to maintain a speed of 24,818.58 Miles/248.1918 Hours = 99.99759 Miles per hour to exactly match the speed of the airplane.

Answer 3

This is a simple geometry problem. You need to compute the total circumference of the flight path of the aircraft at 500 feet above the ground versus the circumference at the earth's surface. Remember that the circle of the aircraft is 1000 feet wider diameter (not 500 feet) because we are computing the flight path diameter is 500 feet wider on one side of the earth and 500 feet wider on the opposite side of the earth.

By the way, the generic formula is:

SpeedG = SpeedA * Circumference / (Circumference + 2 * Height)

Just use common units for Circumference and Height (feet, meters, miles, or kilometers). Units for SpeedG and SpeedA are the same, but don't need to match the other variables.

Diameter of earth = 7,926 miles
Pi = 3.14159
Circumference of earth = 24,902
Feet per mile = 5280
Diameter of earth in feet = 41849280

Circumference of earth surface = 41849280 * 3.14159 = 131473279.5552
Diameter of path of plane = 41849280 + 500 + 500 = 41850280
Circumference of flight path = 41850280 * 3.14159 = 131476421.1452

Answer:
Circumference of earth surface / Circumference of flight path * 100 MPH =
0.99997610529726443885202201753489 * 100 = 99.997 MPH

The ratio of the circumferences will give you the ratio of the speeds. Note that you could simply take the ratios of the diameters because pi is a common factor.

Answer: 99.997 miles per hour (approximately)

However, in reality, the earth's surface is not a perfect sphere and there are very few flat level places on earth to try this out.

Answer 4

zandi has a good point. in order for the car to stay right underneath the plane, the car has to go just slightly slower because the earth is round and the plane is flying on a slightly larger arc.
That being said, it seems to me that the real problem is with understanding that it doesn't matter how high the plane is flying. If the plane is at 35,000ft or 35ft, it can only fly 100mph. so a car would have to go 100mph to stay underneath it.

Answer 5

The same pretty much. Unless there is a wind factor or a certain angle the plane like at a stall angle or a dive. The 500 Feet is probably there to mess with you. Its really no matter. Just make your car go 100 MP/H and your going as fast as the plane.

Answer 6

If the plane flew straight and not follow the earths curvature I guess you would eventualy lose the plane and have to catch up to it by going strait off the earth. If you are going 500 MPH in a car strait ahead on the road then you would still be going 500 MPH if you were 10FT in the air or 500FT in the air. Just make sure when your on the road you have to stop at stop signs and use your turn signals. Then you will have to go faster to catch up. bet you never thought of that one, UH!!!!

Fred M. Hunter
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Answer 7

100 is 100 is 100. if you are doing 100 on the ground, its the same as 100 in the air.

Answer 8

Well it depends on the air friction the ground that your on( rocky cement etc...). Any way why would you want to unless you wanted to throw someone their happy meal they forgot or something then that would be understandable.

Answer 9

100 mph. You have to go the same speed to keep up with it.

Answer 10

if the plane and you are traveling level then you would have to travel 100mph