What is the maximal height a pole vaulter could theoretically jump? Would it ever be possible that a pole vaulter passes the height of 10 meter?

In pole vaulting kinetic energy (= energy of movement) is be transferd into potential energy. The total energy which is the sum of the kinetic and potential energy is conserved at every instant.

At the point that the pole vaulter is going to jump, the total energy E = 1/2 * M * v^2 + M * g * l / 2. Here M is the mass of the pole vaulter (we neglect the mass of the pole with respect to the mass of the pole vaulter), v the speed of the pole vaulter, g the gravitational acceleration constant (9.8 m/s^2) and l the length of the pole vaulter. We assumed that the center of mass of the pole vaulter is halfway his length.

If the pole vaulter is at the highest point, his kinetic energy is about zero, since he almost doesn't move anymore at that point. The total energy E is than equal to the potential energy, E = M * g * h, where h is the height of the jump.

So 1/2 * M * v^2 + 1/2 * M * g * l = M * g * h. From this it follows that h = (l + v^2 / g) / 2.

So a high running speed is of great importance for getting high jump. The maximal reachable running speed can be estimated by realizing that the world's best athletes can run 100 meters in 10 seconds. The pole vaulter will probably not go faster.

So for a jump as high as possible we take v = 10 m/s. Suppose the pole vaulter has a length of 2 meter. Then h = (2 + 10*10 / 9.81) / 2 = 6.1 meter. By pushing yourself from the pole it is possible to put some additional energy into the system. As a result it could be possible to reach somewhat higher than 6.1 meter.

The pole vaulting world record is 6.14 meters by Sergey Bubka from Ukraine. This is nearly a perfect jump, and probably very difficult to beat.

A jump of 10 meters is impossible!