On an essentially frictionless, horizontal ice rink, a skater moving at 5.0 m/s encounters a rough patch that reduces her speed by 40 % due to a friction force that is 22 % of her weight. Use the work-energy theorem to find the length of this rough patch. Express your answer using two significant figures.

Answer: the length of the rough patch is 4.9 m.

Explanation:

The work - energy theorem, in simple words, expresses that the change in the kinetic energy in an object, is equal to the work done on the object by non-conservative forces, like frictional ones.

In the question, we know that the skater is moving at a given speed, and due to the effects of friction, her speed is reduced to 40%.

Added to this, we are told that the friction force is equal to 22% of her weight so we can write the following:

Ff = 0.22 m. g.

Now the work done by this force, is equal to the product of the force times the distance during which the force acted, i. e,. the distance that we are looking for.

So, we can write the following in a brief:

ΔK = Wf ⇒ 1/2 m (vf² - v₀²) = - Ff. x

(The negative sign explains that the friction force always opposes to the relative movement between the two surfaces in contact).

Replacing by the values, and solving for x, we get:

1/2 (2² - 5²) = - 0.22. 9.8. x ⇒ x = 4.9 m