A Sledding Contest. You are in a sledding contest where you...

A Sledding Contest. You are in a sledding contest where you start at a height of $40.0 \mathrm{m}$ above the bottom of a valley and slide down a hill that makes an angle of $25.0^{\circ}$ with respect to the horizontal. When you reach the valley, you immediately climb a second hill that makes an angle of $15.0^{\circ}$ with respect to the horizontal. The winner of the contest will be the contestant who travels the greatest distance up the second hill. You must now choose between using your flat-bottomed plastic sled, or your "Blade Runner," which glides on two steel rails. The hill you will ride down is covered with loose snow. However, the hill you will climb on the other side is a popular sledding hill, and is packed hard and is slick. The two sleds perform very differently on the two surfaces, the plastic one performing better on loose snow, and the Blade Runner doing better on hard-packed snow or ice. The performances of each sled can be quantified in terms of their respective coefficients of kinetic friction on the two surfaces. For the plastic sled: $\mu=0.17$ on loose snow, and $\mu=0.15$ on packed snow or ice. For the Blade Runner, $\mu=0.19$ on loose snow, and $\mu=0.07$ on packed snow or ice. Assuming the two hills are shaped like inclined planes, and neglecting air resistance, (a) how far does each sled make it up the second hill before stopping? (b) Assuming the total mass of the sled plus rider is $55.0 \mathrm{kg}$ in both cases, how much work is done by nonconservative forces (over the total trip) in each case?

Key Concepts

Energy Conservation

This principle states that the total energy in an isolated system remains constant, though it may transform between different forms such as potential energy, kinetic energy, and thermal energy. In problems where friction is present, mechanical energy is not conserved by itself because some of the energy is converted into thermal energy via friction.

Work-Energy Theorem

The work-energy theorem connects the net work done on an object to its change in kinetic energy. When nonconservative forces like friction act on an object, this theorem helps account for the energy lost from the system and is central in determining quantities such as the distance traveled up an incline before coming to rest.

Kinetic Friction

Kinetic friction is the resistive force that acts opposite to the direction of motion when objects slide against each other. It is quantified using a coefficient of kinetic friction and the normal force. This concept is essential for analyzing energy loss in systems where objects move along surfaces with friction.

Inclined Plane Dynamics

Analyzing motion on an inclined plane involves decomposing gravitational forces into components parallel and perpendicular to the plane. This concept is crucial for solving problems involving slopes, as it sets the foundation for determining acceleration, frictional forces, and the effective work done against gravity.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its height in a gravitational field. In energy transformation problems like sledding on hills, this stored energy is converted into kinetic energy and is also partially dissipated by friction as the object moves along the slope.

Transcript

00:02 Okay, let's take a look at this system.

00:05 We are sledding down a slope that is 25 degrees above the horizontal with the height of 40 meters, and we would experience a frictional force of mu 1 going down this slope, and eventually we would go up another slope that is 15 degrees apart above the horizontal, and we would experience a frictional force.

00:35 Fmu2 as we go up this slope and we would eventually stop at the top of this slope.

00:47 Now, slope number one would have loose snow with a length of l1 and slope number two would have a slope with ice on it with a length of l2, which would we need to find out.

01:08 Okay, so we need to figure out how far up the 15 degree slope can each sled can make it.

01:18 So let's take a look at the plastic one.

01:22 So down the 25 degree slope we would have a frictional constant of b1 equals 0 .17.

01:32 So all we needed to use is the conservation of energy as it goes down this slope.

01:38 So p initial plus ke initial minus the work done by the friction is equal to the k -kinetic energy at the valley and the potential energy at the valley, which would be over here.

02:01 Okay, so initially we would have a kinetic energy of zero because we were stationary on the top of that hill.

02:09 So kinetic energy would be zero.

02:11 And at the bottom of the hill we would have a potential energy of zero because the height would be zero at that valley.

02:20 So we would simplify to this with mgh minus the force acted by the friction times the distance it travels along the slope.

02:39 And to figure out what the frictional force is, we know it is mu 1 and times cosine 25 degrees because that would be the force that acts on us as we go down the slope which would be parallel to the slope that we traversed.

03:07 Now figuring out l1 in terms of the height and some trig function.

03:16 So l1 would be h over sine 25 degrees because sine is opposite to the hypotenuse, sine is equal to h over l1, and we manipulate that to get l1 itself, and we substitute it to become h over sign of 25 degrees, and we manipulate to simplify it further, and we will get equation 1...

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