A skier with a mass of 63 kg starts from rest and skis down an icy (frictionless) slope that has

A skier with a mass of 63 kg starts from rest and skis down...

A skier with a mass of $63 \mathrm{kg}$ starts from rest and skis down an icy (frictionless) slope that has a length of $50 \mathrm{m}$ at an angle of $32^{\circ}$ with respect to the horizontal. At the bottom of the slope, the path levels out and becomes horizontal, the snow becomes less icy, and the skier begins to slow down, coming to rest in a distance of $140 \mathrm{m}$ along the horizontal path. (a) What is the speed of the skier at the bottom of the slope? (b) What is the coefficient of kinetic friction between the skier and the horizontal surface?

Key Concepts

Work-Energy Principle

The work-energy principle relates the work done on an object to its change in kinetic energy. This principle is especially useful when non-conservative forces, such as friction, are present. The work done by these forces, which is the product of the force and the distance over which it acts, leads to a reduction in kinetic energy, directly linking frictional forces to the deceleration of the object.

Kinetic Friction and Coefficient of Friction

Kinetic friction is the resistive force that acts against the relative motion of two contacting surfaces when they are sliding past each other. The coefficient of kinetic friction is a dimensionless constant that quantifies the ratio of the frictional force to the normal force. It is a crucial parameter in calculating the magnitude of the frictional force and understanding how surfaces interact, particularly when analyzing motion paths that include deceleration due to friction.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position relative to a gravitational field. When an object moves to a lower height, this energy is converted into other forms, such as kinetic energy. In problems involving motion on an incline, the loss of gravitational potential energy corresponds to the increase in kinetic energy, assuming negligible energy losses due to friction.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that, in the absence of non-conservative forces like friction, the total mechanical energy (the sum of potential and kinetic energies) of a system remains constant. For a frictionless or nearly frictionless path, the energy converted from gravitational potential to kinetic energy can be calculated directly, allowing for the determination of the object's speed.

Transcript

00:01 We first want to find the speed at the bottom of the slope and the coefficient of friction between the horizontal path or the horizontal surface and skier.

00:08 And so we can first take the sum of forces in the y direction for the skier on the slope.

00:14 This would be equal to the mass times the acceleration of y direction, giving us zero.

00:19 This would be equal to the force normal minus the y component of the gravitational force.

00:24 The force normal is then found to be equal to m g cosine of theta and so now we can oh my apologies we can uh we we know that the slope is frictionless so we don't have to include a a friction a friction a friction a force here but once she hits this horizontal path we will have to account for friction so first we can then solve for the force normal.

01:00 We can then apply the newton's second law in the x direction.

01:03 This would be the mass times the acceleration in the x direction.

01:06 This will be equal to mg sine of theta.

01:14 This would be, of course, the acceleration.

01:19 This would of course be the acceleration, or rather the component of the gravitational force that is parallel to the ramp.

01:30 And so the mass is canceled out, and we find that the acceleration and the x direction is equaling g sign of theta.

01:36 This would be equal to 9 .8 meters per second squared multiplied by sign of 32 degrees, the angle of the incline.

01:48 And this is giving us 5 .2 meters per second squared.

01:53 So now we have to find then the final speed at which she is traveling when she reaches the end of, the slope this will be solved through kinematics we can say that then velocity final squared equals velocity initial squared plus two times the acceleration in the x direction multiplied by the length of the incline and so we know that the initial velocity is zero so the final velocity would then be equal to the square root of two multiplied by 5 .2 meters per second squared multiplied by 50 meters.

02:34 And we then find that the final velocity is going to be equal to approximately 22 .8 meters per second.

02:42 So this would be the final velocity at the bottom of the slope for part a.

02:47 For part b, we want to find the coefficient of friction between the horizontal path and the horizontal path and the skier...

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