## To determine how high Jane can swing upward, we need to consider the conservation of energy.

1. Firstly, we need to identify the initial and final positions of Jane. The initial position is when she grabs the vine, and the final position is the highest point she reaches while swinging upward.

2. At the initial position, Jane has both kinetic energy (due to her running speed) and gravitational potential energy (due to her height above the ground). At the final position, Jane's kinetic energy is zero, and all her energy is converted into gravitational potential energy.

3. The principle of conservation of energy states that the total energy of a system remains constant. Therefore, the initial energy equals the final energy.

4. The initial energy can be calculated using the kinetic energy formula: Kinetic Energy = (1/2) * mass * velocity^2. As we don't have information about Jane's mass, we can assume a value of 60 kg, which is an average mass for an adult.

Initial Kinetic Energy = (1/2) * mass * velocity^2

= (1/2) * 60 kg * (5.6 m/s)^2

= 470.4 Joules

5. At the final position, Jane's energy is completely converted into gravitational potential energy, which is given by the formula: Gravitational Potential Energy = mass * gravitational acceleration * height.

Final Potential Energy = mass * gravitational acceleration * height

= 60 kg * 9.8 m/s^2 * height

6. Equating the initial and final energies, we have:

Initial Kinetic Energy = Final Potential Energy

470.4 Joules = 60 kg * 9.8 m/s^2 * height

7. Solving for height, we find:

height = (470.4 Joules) / (60 kg * 9.8 m/s^2)

= 0.794 meters

Therefore, Jane can swing upward to a height of approximately 0.794 meters.