A ski trail makes a vertical descent of 78.0 m (as shown in the figure below ). A novice skier, unable to control his speed, skis down this trail and is lucky enough not to hit any trees. If the skier is moving at 12.1 m/s at the bottom of the trail, calculate the total work done by friction and air resistance during the run. The skier's mass is 68.9 kg.

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So far, I've been using
mgh+1/2mv^2=mgh+1/2mv^2 for the beginning and end of the run, so with an initial velocity of 0 and a height of 0 at the end, I get
52667.16=5043.824, the difference of which is 47623.3355J.

However, that is not the answer and I'm not sure how to continue. Any assisstance would be greatly appreciated!

You forgot the friction.

IntialPE=finalKE+frictionwork

As an aside, your use of significant digits is lacking.

We haven't learned how to do the friction work yet.

And I had the answer, but I forgot to make it negative. Thank you though.

And in reply to your aside, I only use significant digits in my answer.

To calculate the total work done by friction and air resistance during the run, you need to consider the work done against these forces.

First, let's calculate the gravitational potential energy at the top and bottom of the run:

Potential energy at the top (initial point): PE1 = mgh = 68.9 kg * 9.8 m/s^2 * 78.0 m

Potential energy at the bottom (final point): PE2 = 0 since the height is 0 at the bottom

Next, let's calculate the kinetic energy at the bottom (final point):

Kinetic energy at the bottom: KE2 = 1/2 * m * v^2 = 0.5 * 68.9 kg * (12.1 m/s)^2

To find the work done against friction and air resistance, we need to find the change in mechanical energy:

Change in mechanical energy = PE2 + KE2 - PE1

Now, let's substitute the values into the equation:

Change in mechanical energy = 0 + [0.5 * 68.9 kg * (12.1 m/s)^2] - [68.9 kg * 9.8 m/s^2 * 78.0 m]

Simplifying this equation will give you the total work done by friction and air resistance during the run.

To calculate the total work done by friction and air resistance during the skier's run, we need to consider the change in mechanical energy.

The initial mechanical energy of the skier is the potential energy, given by mgh, where m is the mass (68.9 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (78.0 m). The initial kinetic energy is zero since the skier starts from rest.

The final mechanical energy of the skier is the sum of the potential energy, given by mgh, and the kinetic energy, given by 1/2mv^2, where v is the velocity at the bottom of the trail (12.1 m/s).

Therefore, the change in mechanical energy is given by:

ΔE = (mgh + 1/2mv^2) - (mgh + 1/2m(0)^2)

Simplifying, we get:

ΔE = (1/2mv^2) - (1/2m(0)^2)
= (1/2m)(v^2 - 0^2)
= (1/2m)(v^2)

Substituting the given values, we have:

ΔE = (1/2)(68.9 kg)(12.1 m/s)^2

Calculating this expression:

ΔE = (1/2)(68.9 kg)(146.41 m^2/s^2)
= 5095.794 J

This change in mechanical energy represents the work done by the friction and air resistance during the skier's run. Therefore, the total work done by friction and air resistance is approximately 5095.794 J.