A puck of mass .30 kg is sliding along an essentially frictionless patch of ice at 6.0 m/s. It encounters a rough ice patch with a coefficient of kinetic friction of .10. How long will it take for the puck to stop?

KE=W(fr)

mv₀²/2=F(fr)s=μmgs
s= v₀²/2μg
v=v₀-at
v=0 => a= v₀/t
s=v₀t-at²/2=
= v₀t- v₀t²/2t= v₀t/2
t=2s/v₀=2v₀²/2μgv₀=
=v₀/μg=6/0.1•9.8 =6.12 s

To find the time it takes for the puck to stop, we need to first determine the deceleration caused by the friction force. The friction force can be calculated using the equation:

Friction force = coefficient of kinetic friction * normal force

The normal force acting on the puck is equal to its weight, which can be calculated using the equation:

Weight = mass * acceleration due to gravity

Since the patch of ice is essentially frictionless, there is no vertical acceleration acting on the puck. Therefore, the normal force is equal to the weight.

Next, we can calculate the friction force using the given coefficient of kinetic friction and the normal force.

Friction force = coefficient of kinetic friction * normal force

To calculate the deceleration caused by the friction force, we use Newton's second law of motion, which states that the net force on an object is equal to its mass multiplied by its acceleration.

Net force = mass * acceleration

In this case, the net force is equal to the friction force, and the acceleration is the deceleration since the puck is slowing down.

Finally, we can calculate the time it takes for the puck to stop using the following kinematic equation:

final velocity = initial velocity + acceleration * time

Since the final velocity is zero (as the puck stops), we can solve for time.

Let's plug in the given values into the equations:

Mass of the puck (m) = 0.30 kg
Initial velocity (v0) = 6.0 m/s
Coefficient of kinetic friction (μ) = 0.10
Acceleration due to gravity (g) = 9.8 m/s²

First, we need to calculate the weight:

Weight = mass * acceleration due to gravity
Weight = 0.30 kg * 9.8 m/s²
Weight = 2.94 N

Next, let's calculate the friction force:

Friction force = coefficient of kinetic friction * normal force
Friction force = 0.10 * 2.94 N
Friction force = 0.294 N

Now, let's calculate the deceleration:

Net force = Friction force
mass * acceleration = Friction force
acceleration = Friction force / mass
acceleration = 0.294 N / 0.30 kg
acceleration = 0.98 m/s²

Finally, let's calculate the time it takes for the puck to stop:

final velocity = initial velocity + acceleration * time
0 = 6.0 m/s + (-0.98 m/s²) * time
-6.0 m/s = -0.98 m/s² * time
time = -6.0 m/s / (-0.98 m/s²)
time = 6.12 s

Therefore, it will take approximately 6.12 seconds for the puck to stop.