Question

A boy goes sledding down a long \(30^{\circ}\) slope. The combined weight of the boy and his sled is \(72 \mathrm{lb}\) and the air resistance (in pounds) is numerically equal to twice their velocity (in feet per second). If they started from rest and their velocity at the end of \(5 \mathrm{sec}\) is \(10 \mathrm{ft} / \mathrm{sec}\), what is the coefficient of friction of the sled runners on the snow?

Short answer

The coefficient of friction of the sled runners on the snow is approximately \(0.048\).

Step-by-step solution

Calculate the gravitational and air resistance forces

First, we calculate the force due to gravity acting on the sled. We can do this using the formula: \(F_{gravity} = mg \sin(\theta)\), where \(m\) is the mass of the sled and the boy, \(g\) is the gravitational constant, and \(\theta\) is the angle of the slope. Next, we calculate the air resistance force. We are told that the air resistance force is numerically equal to twice the velocity.

Write Newton's second law for the sled

Now, we can write Newton's second law for the sled: \(F_{net} = ma\), where \(F_{net}\) is the net force acting on the sled, \(m\) is the mass of the sled and the boy, and \(a\) is their acceleration. The net force is the difference between the gravity force and the air resistance force adjusted by the friction force: \(F_{net} = F_{gravity} - F_{air resistance} - F_{friction}\).

Calculate the acceleration of the sled

To calculate the acceleration, we first need to find the final velocity of the sled after 5 seconds (given that the initial velocity is 0 ft/s). We can use the formula for uniformly accelerated motion: \(\Delta v = at\), where \(\Delta v\) is the change in velocity, \(a\) is the acceleration, and \(t\) is time. We are given that the final velocity is 10 ft/s, so we can use this equation to find the acceleration of the sled.

Calculate the friction force

We can now use Newton's second law and our calculated acceleration to determine the friction force acting on the sled: \(F_{friction} = F_{gravity} - F_{air resistance} - ma\).

Find the coefficient of friction

Finally, the friction force can be calculated using the formula: \(F_{friction} = \mu_{k} N\), where \(\mu_{k}\) is the coefficient of kinetic friction and \(N\) is the normal force. We know that the normal force is equal to the gravitational force acting perpendicular to the slope: \(N = mg \cos(\theta)\). By substituting this into the friction force equation, we can solve for the coefficient of friction \(\mu_{k}\).