You've entered a "slow ski race" where the winner is the...

You've entered a "slow ski race" where the winner is the skier who takes the longest time to go down a $15^{\circ}$ slope without ever stopping. You need to choose the best wax to apply to your skis. Red wax has a coefficient of kinetic friction $0.25,$ yellow is 0.20 green is $0.15,$ and blue is $0.10 .$ Having just finished taking physics, you realize that a wax too slippery will cause you to accelerate down the slope and lose the race. But a wax that's too sticky will cause you to stop and be disqualified. You know that a strong headwind will apply a $50 \mathrm{N}$ horizontal force against you as you ski, and you know that your mass is 75 kg. Which wax do you choose?

Key Concepts

Newton's Second Law

This law states that the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. It forms the foundation for analyzing motion by providing a relationship between all the forces acting on an object and the way its speed or direction changes.

Force Decomposition on an Inclined Plane

When an object is on a slope, gravity can be split into two components: one parallel to the incline that drives the object downward, and one perpendicular to the incline that contributes to the normal force. This decomposition is essential when calculating the forces that cause motion or resist motion on an inclined surface.

Kinetic Friction

Kinetic friction is the force that opposes the motion of an object sliding over a surface. It is calculated by multiplying the coefficient of kinetic friction by the normal force. This concept is crucial for understanding how different surfaces (or treatments like waxes) affect the speed and acceleration of a moving object.

Additional External Forces

External forces, such as wind, must be considered as they can add to or counteract the other driving forces on an object. Understanding how to decompose these forces into components aligned with the direction of motion allows for a more accurate analysis of the overall net force acting on the object.

Balancing Forces for Controlled Motion

In racing scenarios or other dynamic systems, maintaining a controlled speed often requires balancing the forces that cause acceleration with those that resist movement. Achieving a near-zero net force can help sustain a constant velocity and prevent unintended acceleration or deceleration, which is vital in optimizing performance under varying conditions.

Transcript

00:01 So in this scenario, we are doing a slow ski race down a 15 degree slope, and we need to make sure that we're not stopping during this race.

00:11 We need to choose the best wax to apply to our skis.

00:15 We've got a few different waxes with different coefficients of friction.

00:19 And having just taken physics, you realize that a wax that is too slippery, so too low a coefficient of friction will cause you to accelerate and lose the race.

00:30 But a wax that's too sicky, so has too high of a coefficient, you will stop and you'll be disqualified.

00:38 So there is a headwind that is opposing your motion as you ski and you know that your mass is 75 kilograms and you're asked to calculate what wax would be the best choice.

00:53 So let's go ahead and do a free body diagram to see what forces are acting on this person.

00:59 And we're going to use our standard coordinate system for these type of incline plane problems.

01:08 So here we go.

01:10 I've got my skier here.

01:12 We've got a normal force that's perpendicular to the slope.

01:16 There's going to be that force from the wind and there's going to be a force of friction as well, opposing the motion.

01:25 And then there's going to be a gravitational force straight down.

01:28 So we're really interested in the x direction here.

01:32 We want to make the acceleration as small as possible, but without slowing us down.

01:42 Because if we slow down, then we're going to stop, right? so we don't want to slow down.

01:52 That's a no -no.

02:00 And so that means that we don't want an acceleration that is less than zero.

02:04 We do want to be moving.

02:15 We want to be accelerating a little bit.

02:19 Ideally, our acceleration would be zero, right? because then we just give ourselves a little kick to get going, and then we wouldn't speed up anymore, and we would be going the slowest possible that we could.

02:32 So ideally, our acceleration would be zero.

02:36 So what i would recommend here is creating a fnetx equation, where the acceleration is zero.

02:46 Using that, we can find out what the ideal friction force is and what the ideal coefficient of friction is...

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