What is the terminal speed for an 80 kg skicr going down a...

Key Concepts

Drag Force

Drag force is the resistive force exerted by a fluid, such as air, on a moving object. It typically depends on parameters including the object's cross-sectional area, the fluid density, the drag coefficient, and the square of the object's speed. Understanding drag is crucial for determining when the gravitational force component is balanced by resistance, resulting in terminal velocity.

Equilibrium of Forces

In dynamics, equilibrium occurs when the sum of forces acting on an object is zero. For an object reaching terminal velocity, the forward force (e.g., gravitational component along an incline) is exactly balanced by the resistive forces (such as drag), leading to zero net acceleration. This concept is fundamental to solving problems involving steady-state motion.

Terminal Velocity

Terminal velocity is the constant speed attained by an object moving under the influence of gravity when the net force acting on it becomes zero due to the balance between the driving force (gravity) and the opposing drag force. It is reached when acceleration ceases, and the object continues to move at a steady speed.

Force Components on an Inclined Plane

When an object moves on an inclined plane, the force of gravity can be decomposed into two components: one perpendicular to the surface (which influences the normal force) and one parallel to the surface (which propels the object downhill). This decomposition is essential for analyzing motion on slopes in any similar physical situation.

Transcript

00:01 So in this problem, we're asked to find the terminal speed of an 80 -kilogram skier going down a 40 -degree slope on wooden skis, and we are to assume the skier is 1 .8 meters tall and 0 .4 meters wide.

00:16 Now, in order to do this, we're going to need to create a force balance equation from which we will be able to calculate the terminal speed.

00:25 So in order to do that, let's start off with a free body diagram.

00:29 So let's take this as our incline.

00:39 And this will be our skier.

00:41 There will be a force of gravity pointed straight down.

00:45 A normal force perpendicular to the surface.

00:50 And then we're going to have a drag force as well as a friction force pointing up the slope.

01:02 So these two forces are sort of hindering the motion here.

01:08 Okay.

01:09 So now that we have that, we can go ahead and analyze the motion in the x direction and hopefully we'll be able to find the terminal speed.

01:19 So let's go ahead and calculate the f net x equation.

01:26 And just for reference, we are taking the perpendicular to the incline direction as y and then parallel to the incline will be x.

01:36 So the only force that is in the positive x direction would be the x component of gravity.

01:42 That's so it's pulling this gear down.

01:44 And then opposing that, we have the friction force, as well as the drag force.

01:51 So let's substitute in for a few things here.

01:54 So fgx will be equal to fg instead of fg, let's write mg sign of 40 degrees.

02:08 And if you need a little reminder of how we get that, i will draw the triangle over here.

02:14 So if this is fg, then we've got components.

02:17 With f y and fgy or fg x and this angle at the top of the triangle will always be the same as the incline so that's 40 degrees then the friction force will be mu k fn and the drag force we know will be one quarter a v squared now let's make this the terminal speed and we know that if the skier is moving with the terminal speed then the acceleration is going to be equal to 0, right? the terminal speed is the speed at which all of the forces will balance one another.

02:59 Now what's fn? we need to kind of analyze the y direction for fn...

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