I'm assuming you're imagining an idealized scenario where the earth is rigid and so doesn't distort when it rotates faster (i.e. you want the force of gravity to stay constant). And, from your picture, it appears you have the equator in mind.

Yes, you would be able to jump higher. Rotation seems to muck with people's intuition for non-inertial motion. If, instead of centripetal acceleration, you imagine an elevator accelerating downward at the same acceleration, the result is easier to picture. Meanwhile, for a sufficiently small elevator, the effect of gravity is equivalent to accelerating upwards at ##9.8 m/s^2##. Clearly your net acceleration will change if you downward acceleration vector changes (corresponding to a greater centripetal acceleration), and this net acceleration is the one you plug into the kinematic equations. You might find it more enlightening to examine the whole situation from an inertial reference frame. Of course, for the effect to actually be noticeable the earth would have to rotate so fast that the Coriolis effect would also probably be apparent. And if you're not at the equator, the centrifugal and gravitational forces aren't parallel.

Your friend is misunderstanding fictitious forces. They are properties of non-inertial reference frames; not the objects within the reference frame.

To illustrate the point even better, it may help your friend to realize that at sufficiently high rotation speed, the velocity of points on the earth's surface would correspond to the orbital velocity for something at a distance of the earth's radius from its centre. That is, you would be completely weightless like the astronauts in the ISS. That you would be able to jump higher in that extreme scenario shouldn't be very controversial.

^This was the point I was going to make. Geosynchronous orbit is possible at 22,000 miles above the surface, but the height of this orbit depends on the speed you want to rotate at. Orbital mechanics says that if you want to orbit the Earth faster, you must drop into a lower orbit. Consider if the Earth's rotation rate increased by 10%, then a geosynchronous orbit would be at a lower altitude, because you need to be closer to the surface to speed up in order to stay above the same point on the equator.

Well, continue this reasoning to the point where the Earth rotates so fast that a geostationary orbit occurs at 50cm above sea level? You could jump 50cm high and never land again!

The resulting orbit cannot be circular. You start with a vertical component.

If the resulting orbit were closed (elliptical) then it would obviously intersect with the surface of the earth. That only leaves parabolic and hyperbolic trajectories. So the relevant question is whether a vertical jump could result in attaining escape velocity.

Since your starting velocity is equal to orbital velocity (~8 km/sec) and the needed velocity is escape velocity (~11 km/sec), that's a 3 km/sec delta-v requirement.

Worse, that 3 km/sec would be in the direction of rotation. If you wanted to do it with a vertical jump you'd need 8 km/sec of delta-v to get a resultant of 11 km/sec.

That's a heck of a jump, even for Wile E. Coyote wearing Acme rocket shoes.

Yes, although it is interesting to consider climbing up a stepladder.... You could float off when you reached the geosynchronous height.... of course a number of unreasonable stresses and accelerations would be involved.

Do you mean that, in case the spacecraft would catch fire while still in the launch base, they would spin up the Earth to extinguish it with the sea waters?

I would hope that he did not mean that at all. I also hope that none of the discussion on this post as been taken seriously either. Objects on, or at, the surface of this planet, are subject to Gravity. When they begin to move, they are still being acted on by this force. When they are at a stable orbit, they are subject to this force. It is this force that accounts for the circular nature of these objects. The force of Gravity, the acceleration of, creates a centripetal force that accounts for the circular orbits.

I didn't, but I find that interpretation very humourous.
Unlike yours, which is just wrong. Equatorial launches are desired to use the speed of the Earth's rotation as a "springboard" to the rocketry.

What's wrong is the suggestion that nothing in this thread is worth reading. While the initial question is misguided, it does have a valid basis. I can see why someone who read about the choice of Florida for NASA launches and Brazil for private sector ones might wonder if the principle "scales down" to people.