A girl coasts down a hill on a sled, reaching the bottom with a speed of 8.6 m/s. The coefficient of friction between the sled's runners and the snow is 0.050, and the girl and sled together weigh 670 N. How far does the sled travel on the surface before coming to rest?

To find the distance the sled travels before coming to rest, we can use the equations of motion.

First, let's calculate the acceleration of the sled using the coefficient of friction and the weight of the girl and sled.

The frictional force can be calculated using the equation:

Frictional force = coefficient of friction * normal force

Normal force = weight of the girl and sled = 670 N

Frictional force = 0.050 * 670 N

Frictional force = 33.5 N

The net force on the sled is the difference between the gravitational force and the frictional force:

Net force = weight of the girl and sled - frictional force

Net force = 670 N - 33.5 N

Net force = 636.5 N

Now, we can use Newton's second law of motion to calculate the acceleration:

Net force = mass * acceleration

Acceleration = Net force / mass

Acceleration = 636.5 N / 670 N (convert weight to mass by dividing by acceleration due to gravity)

Acceleration = 0.949 m/s^2

Next, we can use the equation of motion to find the distance traveled by the sled:

v^2 = u^2 + 2as

where:
v = final velocity (0 m/s as the sled comes to rest)
u = initial velocity (8.6 m/s)
a = acceleration (-0.949 m/s^2)
s = distance traveled

Substituting the given values into the equation, we have:

0^2 = 8.6^2 + 2 * (-0.949) * s

Simplifying the equation:

0 = 73.96 - 1.898s

Rearranging the equation to solve for s:

1.898s = 73.96

s = 73.96 / 1.898

s ≈ 39 meters (rounded to two decimal places)

Therefore, the sled travels approximately 39 meters on the surface before coming to rest.

To find the distance the sled travels before coming to rest, we can use the concept of work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.

Here's how we can approach this problem step by step:

Step 1: Calculate the initial kinetic energy of the sled.
The initial kinetic energy of the sled can be calculated using the formula:
Initial kinetic energy = 0.5 * mass * velocity^2

Since the mass of the sled is not provided, we need to calculate it using the weight of the girl and the sled:
mass = weight / acceleration due to gravity

Given that the weight of the girl and the sled together is 670 N, and the acceleration due to gravity is approximately 9.8 m/s^2, we can find the mass.

mass = 670 N / 9.8 m/s^2

Step 2: Calculate the work done on the sled.
The work done on the sled can be calculated using the formula:
Work done = force * distance

In this case, the force acting against the motion of the sled is the frictional force. The frictional force can be calculated using the formula:
Frictional force = coefficient of friction * normal force

The normal force in this case is equal to the weight of the girl and the sled.

frictional force = coefficient of friction * weight

Step 3: Calculate the distance traveled by the sled.
Since the work done is equal to the change in kinetic energy, we can use the formulas for work done and kinetic energy to solve for distance:
Work done = change in kinetic energy
Work done = Final kinetic energy - Initial kinetic energy

Finally, we can rearrange the equation to solve for distance:
distance = (Final kinetic energy - Initial kinetic energy) / frictional force

Now, we can plug in the values and solve for the distance:
distance = [(0 - 0.5 * mass * velocity^2) / frictional force]

Substituting the calculated values from earlier, we can find the distance the sled travels before coming to rest.

V^2 = Vo^2 + 2a*d

V = 0
Vo = 8.6 m/s.
a = u*g = 0.050*(-9.8) = -0.49 m/s^2
Solve for d.