1. Define the Goal:

We need to find the initial vertical velocity (v₀) required for a person to reach a height of 1.85 meters (center of mass) plus 0.65 meters (crossbar), totaling 2.5 meters.

2. Set Up the Energy Equation:

* Initial Energy: The person starts with kinetic energy (KE) only:

KE = (1/2)mv₀²

* Final Energy: At the highest point, the person has only potential energy (PE):

PE = mgh

Where:

* m = mass of the person

* v₀ = initial vertical velocity

* g = acceleration due to gravity (9.8 m/s²)

* h = total height (2.5 meters)

3. Apply Conservation of Energy:

Since energy is conserved, the initial kinetic energy must equal the final potential energy:

(1/2)mv₀² = mgh

4. Solve for the Initial Velocity (v₀):

* Cancel out the mass (m) on both sides.

* Rearrange the equation to solve for v₀:

v₀² = 2gh

v₀ = √(2gh)

5. Calculate the Initial Velocity:

* Substitute the values:

v₀ = √(2 * 9.8 m/s² * 2.5 m)

v₀ ≈ 7.0 m/s

Therefore, the person must leave the ground with a minimum speed of approximately 7.0 meters per second to clear the crossbar.