(II) A 75-kg snowboarder has an initial velocity of 5.0 m / s at the top of a 28^∘ incline (Fig.

(II) A 75-kg snowboarder has an initial velocity of 5.0 m /...

(II) $\mathrm{A}$ 75-kg snowboarder has an initial velocity of 5.0 $\mathrm{m} / \mathrm{s}$ at the top of a $28^{\circ}$ incline (Fig. $36 ) .$ After sliding down the 110 -m long incline (on which the coefficient of kinetic friction is $\mu_{k}=0.18$ ), the snowboarder has attained a velocity $v .$ The snowboarder then slides along a flat surface (on which $\mu_{k}=0.15$ and comes to rest after a distance $x .$ Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface. Then use these accelerations to determine $x .$

(II) $\mathrm{A}$ 75-kg snowboarder has an initial velocity of
5.0 $\mathrm{m} / \mathrm{s}$ at the top of a $28^{\circ}$ incline (Fig. $36 ) .$ After sliding
down the 110 -m long incline (on which the coefficient of
kinetic friction is $\mu_{k}=0.18$ ), the snowboarder has
attained a velocity $v .$ The snowboarder then slides along
a flat surface (on which $\mu_{k}=0.15$ and comes to rest
after a distance $x .$ Use Newton's second law to find the
snowboarder's acceleration while on the incline and
while on the flat surface. Then use these accelerations to
determine $x .$

Key Concepts

Newton's Second Law

Newton's second law explains that the net force acting on an object is equal to the product of its mass and acceleration. This fundamental concept allows us to analyze the motion of objects by accounting for all forces, including gravity and friction, and is essential for setting up equations that describe acceleration in various scenarios.

Kinetic Friction

Kinetic friction is the resistive force that opposes the motion of an object sliding over a surface. Determined by the coefficient of kinetic friction and the normal force, it plays a critical role in dynamics problems by affecting the net force and, consequently, the acceleration of moving objects.

Inclined Plane Dynamics

Inclined plane dynamics involves breaking down gravitational force into components parallel and perpendicular to the surface of the incline. This decomposition is key to understanding how forces interact on an inclined surface, determining the net force that produces acceleration along the slope.

Kinematics of Uniform Acceleration

Kinematics of uniform acceleration deals with the relationships among displacement, velocity, acceleration, and time when acceleration is constant. Utilizing these kinematic equations is essential for predicting the motion of objects under constant acceleration, such as when friction and gravity result in uniform changes in velocity.

Transcript

00:01 So let's draw the pre -board diagram of the snowboarder when he's on the slope.

00:09 So if that's the snowboarder, we see that we have m .g.

00:16 That's acting downwards.

00:17 And then there is the normal force which is acting vertically, which is acting perpendicular to the slope.

00:24 Let's call it fn.

00:26 Then there's friction acting along the slope in the upward direction.

00:35 We call it f of f r and for our convenience let's actually assume that the perpendicular direction to the slope as positive y and down the slope as positive x so and if the slope makes angle theta with the horizontal using trick we can say that that angle theta must be equal to uh that that angle must be equal to the angle between m g and y -axis so using newton's second law we can write down two separate equations one in the x direction one in the wide direction so in the wide direction we have fn minus mg cosine theta is equal to zero which gives us fn is equal to mg cosine theta and in the verge of skidding we say that force of friction is equal to static coefficient of static friction times f of n where f of n is already calculated here so we can use these two pieces of information in our second part of the equation where we solved for f x so f x is equal to m g cosine theta minus f of fr which is equal to m a where m a so m g cosine theta is basically the component of mg along x axis and fr is directed in the opposite direction that's why you have a negative sign here so again as we said that we can combine fn and fr together in this piece and we see that finally we have m g cosine theta minus mu s m g so that's going to be sign theta actually because along extraction it's sign so that's sign theta similarly this will be sign as well so mg sign theta minus mu s mg cosine theta is equal to m times a and from here uh this a is the acceleration in the slope so a slope is equal to g of sine theta minus mu k one f of n so that becomes 9 .8 meter per second squared times sine theta which is sine 28 degrees minus 0 .18 cosine 28 degrees which is equal to 3 .043 meter per second square.

03:57 So here's i did one mistake here while calculating the friction of course it's not the static friction but it's the kinetic friction that we're considering because the snowboard is moving.

04:09 So when when an object is moving we consider the kinetic friction, kinetic coefficient of kinetic friction not the coefficient of static friction.

04:20 So that's why we have mu k1 and um mew k1 is denoting that that's the coefficient of kinetic friction in the slope.

04:30 Now, we can do the same thing for the second part where the snowboarder is on a flat surface.

04:37 So if we do so, we see that if the snowboarder moves on the right, the friction force is acting on the left...

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