Question

A man is pushing a loaded sled across a level field of ice at the constant speed of \(10 \mathrm{ft} / \mathrm{sec}\). When the man is halfway across the ice field, he stops pushing and lets the loaded sled continue on. The combined weight of the sled and its load is \(80 \mathrm{lb}\); the air resistance (in pounds) is numerically equal to \(\frac{3}{4} v\), where \(v\) is the velocity of the sled (in feet per second); and the coefficient of friction of the runners on the ice is \(0.04\). How far will the sled continue to move after the man stops pushing?

Short answer

The sled will continue to move approximately 28.9 feet after the man stops pushing.

Step-by-step solution

Identify the forces acting on the sled

When the man stops pushing, there are two forces acting on the sled: air resistance and the friction between the sled and the ice. We are given the air resistance in terms of velocity (v) as \(0.75v\) lbs and the coefficient of friction as 0.04. The normal force is equal to the weight of the sled, which is 80 lbs. The frictional force can be found using the formula: Frictional Force = Friction Coefficient * Normal Force.

Calculate the frictional force

Now, we will find the frictional force acting on the sled. Use the formula mentioned above and plug in the given values: Frictional Force = 0.04 * 80 = 3.2 lbs

Set up the equation for acceleration

We know that air resistance and frictional force are acting against the sled's motion. We also know that the sled's velocity decreases due to these forces, so there is negative acceleration. Thus, we can set up an equation using Newton's second law: Net Force = Mass * Acceleration Negative net force = air resistance + frictional force We also know that: \(F = ma\) So, \(-ma = 0.75v + 3.2\) Since the sled has a constant speed of 10 ft/s when the man stops pushing, let's plug this into the equation: \(-ma = 0.75(10) + 3.2\)

Convert weight to mass

In order to find the acceleration, we need to first find the mass of the sled instead of its weight. The sled's weight is 80 lbs. To convert weight to mass, divide by 32.17 ft/s², which is the gravitational acceleration: Mass = Weight / Gravitational Acceleration Mass = 80 / 32.17 = 2.486 kg (approximately)

Calculate the acceleration

Now we can plug in the mass (2.486 kg) into the equation from Step 3 to find the acceleration: \(-2.486a = 7.5 + 3.2\) Divide by -2.486 to get the acceleration: \(a = -4.3 / 2.486 \approx -1.73 ft/s^2\)

Calculate the distance using kinematics

We now know the acceleration (-1.73 ft/s²) and initial velocity (10 ft/s). We will use the kinematic equation to calculate the distance when the sled comes to a stop (final velocity = 0): \(v_f^2 = v_i^2 + 2ad\) where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration, and \(d\) is the distance. Since final velocity is 0, the equation becomes: \(0 = 10^2 + 2(-1.73)d\) Solve for distance: \(d = -100 / -3.46 \approx 28.9 ft\) The sled will continue to move approximately 28.9 feet after the man stops pushing.