A 6.7 kg package starts from rest and slides down a mail chute that is 50 m long and has a vertical drop of 10m. If the velocity of the package at the bottom of the chute is 4m/s what is the average frictional force on the package as i slides down the chute?

To find the average frictional force on the package, we need to consider the energy changes involved in its motion.

1. First, let's find the gravitational potential energy (GPE) of the package at the top of the chute and at the bottom. The GPE formula is given by:
GPE = mass * gravity * height

At the top of the chute:
GPE_top = (6.7 kg) * (9.8 m/s²) * (10 m)

At the bottom of the chute, the potential energy is converted entirely into kinetic energy:
GPE_bottom = 0

2. The total mechanical energy of the package is given by the sum of its kinetic energy (KE) and potential energy (PE). The formula is:
Total mechanical energy = KE + PE

At the top of the chute, the package has no kinetic energy:
Total mechanical energy_top = GPE_top + KE_top
Total mechanical energy_top = GPE_top + 0

At the bottom of the chute, the package has only kinetic energy:
Total mechanical energy_bottom = 0 + KE_bottom

3. Since mechanical energy is conserved (ignoring friction), the total mechanical energy at the top is equal to the total mechanical energy at the bottom:
Total mechanical energy_top = Total mechanical energy_bottom

GPE_top = 0 + KE_bottom
GPE_top = 0.5 * mass * velocity²

Substituting the given values:
(6.7 kg) * (9.8 m/s²) * (10 m) = 0.5 * (6.7 kg) * (4 m/s)²

Solving this equation will give us the value of KE_bottom, which represents the kinetic energy of the package at the bottom of the chute.

4. Once we know the kinetic energy at the bottom, we can calculate the corresponding frictional force. The equation for kinetic energy is:
KE = 0.5 * mass * velocity²

Rearranging the equation to find the force:
Force = (0.5 * mass * velocity²) / distance

Substituting the known values:
Force = (0.5 * 6.7 kg * (4 m/s)²) / 50 m

By substituting the values into the equation and calculating, we'll obtain the average frictional force on the package as it slides down the chute.