On a frozen pond, a 10 kg sled is given a kick that imparts to it an initial speed of 6.2 m/s. The coefficient of kinetic friction between sled and ice is 0.14. Find the distance the sled moves before coming to rest. The acceleration of gravity is 9.8 m/s^2.

Answer in units of m.

To find the distance the sled moves before coming to rest, we can use the equation of motion:

vf^2 = vi^2 + 2ad

Where:
- vf is the final velocity (which is 0 m/s since the sled comes to rest)
- vi is the initial velocity (6.2 m/s)
- a is the acceleration (caused by the friction force)
- d is the distance

First, let's find the acceleration using the friction force:

friction force = coefficient of kinetic friction * normal force

The normal force is equal to the weight of the sled:

normal force = mass * gravity

Substituting the given values:

normal force = 10 kg * 9.8 m/s^2

Next, we can calculate the friction force:

friction force = 0.14 * (10 kg * 9.8 m/s^2)

Now, we can calculate the acceleration:

acceleration = friction force / mass

Substituting the known values:

acceleration = (0.14 * (10 kg * 9.8 m/s^2)) / 10 kg

Finally, we can find the distance:

0 = (6.2 m/s)^2 + 2 * acceleration * d

Solving for d:

d = -((6.2 m/s)^2) / (2 * acceleration)

Now we can substitute the values and calculate:

d = -((6.2 m/s)^2) / (2 * (0.14 * (10 kg * 9.8 m/s^2)) / 10 kg)

Calculating this expression gives us the distance d.

To find the distance the sled moves before coming to rest, we'll first calculate the frictional force acting on the sled using the coefficient of kinetic friction. Then we can use Newton's second law of motion to find the acceleration of the sled, and finally, we can use the equations of motion to determine the distance traveled.

1. Calculate the frictional force:
The coefficient of kinetic friction (μk) is given as 0.14.
The normal force (N) on the sled is equal to the weight of the sled, given by N = m * g, where m is the mass of the sled and g is the acceleration due to gravity. In this case, m = 10 kg and g = 9.8 m/s^2.
So, N = 10 kg * 9.8 m/s^2 = 98 N.

To calculate the frictional force (f), use the equation f = μk * N:
f = 0.14 * 98 N = 13.72 N.

2. Calculate the acceleration of the sled:
The net force acting on the sled is the difference between the kick force and the frictional force. Since the sled is coming to rest, the net force is equal to the frictional force.
Using Newton's second law of motion, F = m * a, where F is the net force and a is the acceleration:
13.72 N = 10 kg * a.
Simplifying the equation, we have a = 1.372 m/s^2.

3. Calculate the distance traveled:
To find the distance traveled (d), we can use the equation of motion:
v^2 = u^2 + 2 * a * d,
where v is the final velocity (0 m/s), u is the initial velocity (6.2 m/s), a is the acceleration (1.372 m/s^2), and d is the distance traveled.

Rearranging the equation, we have d = (v^2 - u^2) / (2 * a).
Therefore, d = (0^2 - (6.2 m/s)^2) / (2 * 1.372 m/s^2).

Calculating the result, we find d = (-38.44) / 2.744 ≈ -14.03 m.

However, distance cannot be negative in this context. So, the sled does not move backward; it only travels in the opposite direction to its initial motion.

Hence, the distance the sled moves before coming to rest is approximately 14.03 meters.

normal force = m g = 98.1

so
friction force = .14*98.1

friction force * d = (1/2) m v^2

.14*9.81 * d = 5 (6.2)^2
solve for d