A man dug a hole straight down into the earth and surfaced at the other side of the earth. If he were to jump?

Question

Ok Genius people.. here is the Question for You:) don't work hard:)

A man dug a hole straight down into the earth and surfaced at the other side of the earth. If he were to jump into the hole, how far down would he fall?

Answer 1

Well it depends on your assumptions. If you ignore air-resistance and the inevitable burning death (which I suppose you'd have to if he managed to dig the hole), then he would fall right to the other side. Interestingly, he would move in simple harmonic motion, because when inside a body of mass the gravitational force acting on you from the body is proportional to the distance from the centre of mass. You'd also have to assume the Earth was a perfect sphere and ignore the gravitational effects of any other bodies, but basically he would go right to the other side.
So you could model his displacement from the centre against time as a cosine graph, and his velocity as a sine wave (i.e. his displacement is maximum at t=0, his velocity is zero at t=0, the other way around at the centre of mass.

So with a little knowledge of SHM, you wouldn't need much calculus (not much difficult calculus anyway) to find his speed at any given point. The radius of the earth is 6.4*10^6 m.
Given that his weight is proportional to the distance from the centre, and that it is mg N at the surface, we can say that:
W= -kx =mg (at the surface, there's a minus sign because the force is in the opposite direction to the displacement vector).
Using the values of x=6.4*10^6 and g=9.8m/s^2, we have that k=m*1.53*10^(-6) therefore k/m = 1.53*10^(-6)
At that point g is the acceleration, so we can say that at any point:
-kx=ma. So:
a= -kx/m. This means the angular frequency is [sqrt(k/m)].
this is equal to sqrt(1.53*10^-6)= 1.24*10^(-3)
So to get from one side to the other it would take him
2pi/1.24*10^(-3) = 5080 seconds = 85 hours.
His fastest speed (i.e. speed at the centre) woould be:
1.24*10^(-3) * 6.4*10^6 = 7936m/s = 18,000 miles per hour.
So it would take him 3-and-a-half full days to reach the other side and in the middle he'd be going at 18,000 mph.


However all of these assumptions are bogus, and even if you keep the one about him not dying, there would be massive damping effects from air resistance and he would eventually get stuck in the middle.

Answer 2

Neglecting air resistance, he would fall all the way through, since as he passes the center, he would begin to decelerate, rather than accelerate due to gravity.

If you factor in air resistance, he would fall almost all the way through and then fall back toward the starting point, falling almost as far toward that side. This would continue with him falling less with each pass until he came to rest in the center of the earth.

Answer 3

Assuming no air friction, all the way to the opposite side. In reality, he'd fry digging the hole. He would accelerate all the way to the center of the earth and then start to decelerate until he got to the other side.

Answer 4

He would fall to the earths core. That is where the gravitational force comes from. He would probably go past it and come back until he came to a complete stop at the center of the earth.

P.S. I don't think that this is even physically possible (to dig/drill a hole through the entire earth).

Answer 5

first off... at some point down the "rock" becomes molten... also the pressure goes up so much once you get past he "crust" into the mantle... I would say... about 20 KM... because the hole would colapse... if you made it past the upper mantle you would burn up way before you got to the center... as the temp goes WAY up... lets say at 1000 degrees celcius you would burn up in a few seconds... the center of the earth is 5000 degrees + depends on who you believe.. but it is super duper hot...

So you wouldn't fall to far before you burned to a crisp and nothing was left to fall...

Answer 6

If there was no wind resistance he would fall all the way through the earth to China but since there is wind resistance he would stop somewhere just past the middle and then fall back towards the middle again.

Answer 7

I have wondered about this question for some time. A part of me thinks that he will fall all the way, but have a theoretical velocity of 0 at the end of his fall at the other end of the earth.

Answer 8

actually... assuming he did not scrape against the sides... he would keep going back and forth between sides in a sort of perpetual motion machine... this is also if there was no air friction.

and yes.. perpetual motion is possible: orbiting planets lol

most people dont consider this a problem, but it is, and thats where alot of alternative veiws such as the expanding matter view of the universe come from.... in that view the guy would simply stop at the center.

Answer 9

To the Earth's center

Answer 10

The man will reach the other end of the diameter
From surface to center, the man will gain terrific speed and then the speed will decrease to zero when he reaches the other end if the man's initial speed is zero.
For details, you need integration