a slide loving pig slides down a certain 44° slide in twice the time it would take to slide down a frictionless 44° slide. What is the coefficient of kinetic friction between the pig and the slide?

The force of friction on the slide is

mg*mu*cosTheta, and the component of gravity down the slide is mgsinTheta

F=ma=m(vf/time) but vf with friction is 1/2 that of no friction (why? average velocity is 1/2 that of no friction).

so, with friction...
mgSinTheta-mg*mu*Costheta=m Vf/time
=m 1/2 V'/2 t' where the ' means no friction.
g sinTheta-g*mu*cosTheta=V'/4t'

now without friction..
mg sinTheta -nofriction=m v'/t'
or v'/t'=g sinTheta
now back into the friction equation.
g sinTheta-mu*g*CosTheta=g sinTheta/4
dividing both sides by g cosTheta
tanTheta-mu=tantheta /4
mu=3/4 tanTheta

check that.

To find the coefficient of kinetic friction between the pig and the slide, we can start by understanding the forces acting on the pig as it slides down the slide.

In the first scenario, where the slide is frictionless, the only force acting on the pig is its weight (mg), where m is the mass of the pig and g is the acceleration due to gravity. The pig slides down the slide at a constant acceleration due to gravity, which is approximately 9.8 m/s^2.

In the second scenario, where there is a frictional force between the pig and the slide, there are two forces acting on the pig:

1. The weight of the pig (mg) acting vertically downward.
2. The frictional force (Fk) acting parallel to the slide in the opposite direction of motion.

Since the pig slides down the slide in twice the time it would take on a frictionless slide, it means that the frictional force reduces the acceleration of the pig.

We can use the following formula to calculate the acceleration in both scenarios:

1. For the frictionless slide, the acceleration (a1) can be calculated as:
a1 = g = 9.8 m/s^2

2. For the slide with friction, the acceleration (a2) can be calculated using the equation of motion, where t is the time taken to slide down the slide:
a2 = 2d / t^2

In both cases, d is the length of the slide and t is the time taken to slide down the slide.

Since the angle of inclination (θ) is given as 44°, we can find the length of the slide (d) using the equation:
d = h / sin(θ)

Where h is the vertical height of the slide.

Now, we can compare the two accelerations (a1 and a2) to find the coefficient of kinetic friction (μk) using the equation:
μk = (a1 - a2) / g

Substituting the values and solving the equation will give us the coefficient of kinetic friction between the pig and the slide.