# A girl coasts down a hill on a sled, reaching a level surface at the bottom with a speed of 6.7 m/s. The coefficient of kinetic friction between runners and snow is 0.038, and the girl and sled together weigh 603 N. The acceleration of gravity is 9.8 m/s2 .

How far does the sled travel on the level surface before coming to a rest?

## To find the distance the sled travels on the level surface before coming to a rest, we need to use the concept of work and energy.

The work done by the force of kinetic friction will eventually bring the sled to a stop. The work done is equal to the change in kinetic energy. Let's break down the steps to find the distance.

Step 1: Find the initial kinetic energy

The initial kinetic energy (K1) of the sled can be determined using the formula:

K1 = 1/2 * m * v^2

Given:

Mass of the girl and sled (m) = 603 N / 9.8 m/s^2 (weight divided by acceleration due to gravity)

Initial velocity (v) = 6.7 m/s

Plugging in the values:

K1 = 1/2 * (603 N / 9.8 m/s^2) * (6.7 m/s)^2

Step 2: Find the work done by the force of kinetic friction

The work done by the force of kinetic friction can be calculated using the formula:

Work = force of kinetic friction * distance

Given:

Coefficient of kinetic friction (µ) = 0.038

The force of kinetic friction (F) can be calculated using the formula:

F = µ * normal force

The normal force (N) is equal to the weight of the girl and sled, which is 603 N.

Plugging in the values:

F = 0.038 * 603 N

Now, we need to find the distance (d) that the sled travels.

Step 3: Use the work-energy principle

According to the work-energy principle, the work done by the force of kinetic friction is equal to the change in kinetic energy:

Work = ∆K = K1 - K2

Since the sled comes to rest, K2 (final kinetic energy) is equal to zero.

Setting up the equation:

Work = 0 - K1

Step 4: Find the distance (d)

Using the formula for work:

Work = force of kinetic friction * distance

Plugging in the values:

0.038 * 603 N * d = -K1

Rearranging the equation to solve for d:

d = -K1 / (0.038 * 603 N)

Now, calculate the distance (d) by substituting the value of K1 into the equation.

Note: Since the negative sign indicates that the force and displacement are in opposite directions, we can ignore it here as we are only interested in the magnitude of the distance.

Once you calculate the value, you will have the distance the sled travels on the level surface before coming to rest.