Final answer:
To calculate the average force exerted on Lana by the mat, one must use the work-energy principle to find the change in her kinetic energy and relate that to the work done by the mat over the distance of compression.
Step-by-step explanation:
To determine the average force the mat exerted on Lana during impact, we will use the work-energy principle, which states that the work done by forces on an object equals the change in its kinetic energy.
First, we find the potential energy at the peak height and then at the height of 1.0 m when she makes contact with the mat:
- Potential energy at 5.90 m (PEtop): PE = mgh = (70 kg)(9.8 m/s2)(5.90 m)
- Potential energy at 1.0 m (PEcontact): PE = mgh = (70 kg)(9.8 m/s2)(1.0 m)
The change in potential energy as she falls to the mat is thus PEtop - PEcontact, which is the kinetic energy she has upon contact with the mat since she starts at rest from the peak height. This kinetic energy is the work done on her by her weight to bring her from rest to her velocity at contact.
The work done on her to stop her (which is the work done by the mat) is equal to the negative of her kinetic energy at contact since she comes to rest:
Work = Change in kinetic energy = 0 - (PEtop - PEcontact)
The work done by the mat is also equal to the average force exerted by the mat times the distance over which this force is applied (which is the compression of the mat, or 1.0 m - 0.5 m). Setting these equal gives us:
Average force * Compression distance = -(PEtop - PEcontact)
We then solve for the average force:
Average force = -(PEtop - PEcontact) / Compression distance
Substitute the values for potential energy and compression distance to find the average force.