A skier leaves the ramp of a skijump with a velocity of 10.0...

Key Concepts

Projectile Motion

Projectile motion deals with objects moving under the influence of gravity in two dimensions. It involves analyzing the horizontal and vertical components of motion separately, where the horizontal motion is constant and the vertical motion is accelerated due to gravity. This concept is crucial for solving problems that determine the time of flight, range, and displacement of a projectile.

Kinematics on an Inclined Plane

Kinematics on an inclined plane involves analyzing the motion of an object relative to a surface that is not horizontal. In projectiles landing on an inclined surface, the kinematic equations must take into account the geometry of the landing plane to correctly determine the point of impact, as the effective vertical displacement is measured perpendicular to the inclined surface.

Velocity Decomposition

Velocity decomposition is the process of breaking down a vector into its horizontal and vertical components using trigonometric functions. This concept is essential when analyzing projectile motion, as it allows the separate treatment of the constant horizontal component and the vertically accelerated component due to gravity.

Air Resistance Effects

Air resistance introduces a drag force that opposes the motion of a projectile, reducing its overall range and altering its trajectory. Unlike gravity, which consistently accelerates the object, air resistance depends on the object's velocity, shape, and size, causing both horizontal and vertical components of motion to change non-linearly over time.

Aerodynamic Posture

Aerodynamic posture refers to the orientation and shape an object adopts to minimize drag during motion through the air. In the context of skiers or jumpers, positioning the body in the shape of an airfoil reduces air resistance, which allows for a longer flight distance by preserving more of the initial forward velocity.

Transcript

00:03 Okay, so for this question, it wants us in the first part to find the distance from the ramp to where the jumper lands.

00:10 So that's part a.

00:11 So what we're going to need to do to start out is we're going to need to find the y and x components of the velocity, right when the ski jumper jumps.

00:21 And so we can find that using the angle they've given us and the initial velocity.

00:26 So this is just a little bit of trigonometry.

00:29 We'll say that 10, the initial velocity, is the hypotenuse of this little triangle right here.

00:35 I've kind of drawn it down here.

00:38 So we have this little triangle, the angle, and the hypotenuse.

00:42 So we're going to need to find the y components right here and the x components right here.

00:51 So let's find the y component first.

00:54 So if we have the hypotenuse and we're looking for the opposite side, opposite from the angle, we're going to need to use the sign.

01:07 So we'll do the velocity in the x, or in the y direction, sorry, in the y direction equals the hypotenuse times sign of the angle.

01:24 And that's, we'll plug that into our calculators and make sure your calculators are in degrees rather than radiance for this one.

01:31 So if we plug that into our calculator, we get the y component to be positive 2 .59 meters per second.

01:44 All right.

01:45 Now let's do the x direction.

01:47 So v of x, we're looking for the adjacent side of the triangle.

01:51 So we're going to use cosine this time.

01:52 So 10 times cosine 15 degrees equals positive 9 .66.

02:09 Meters per second.

02:11 So those are the initial components of velocity that we need.

02:15 So in order to find how far the ski jumper goes, we're going to need to find how long he's in the air.

02:22 And we'll do that using the velocity in the y direction.

02:27 So that velocity is pointing upwards.

02:30 So what he's going to do is going to go upwards until gravity overcomes his initial velocity and then you'll fall back down to the earth.

02:38 So what we will do, is we'll find how long it takes for him to reach the peak of his jump, and then we'll multiply that time by two, because he'll go up, then come back down in the same amount of time.

02:52 So we can do this using our kinematic equations.

03:07 So the kinematic equation that we're going to use will be the most simple one.

03:21 It'll be v final equals v initial plus acceleration times time.

03:32 So we know that the acceleration of gravity is 9 .8 meters per second squared.

03:39 And because we're saying the up is in the positive direction, gravity will be in the negative direction, which is down.

03:46 So it will be negative 9 .8.

03:49 So to find time, that's what we're looking for, is this time right here.

03:54 We're going to plug some numbers in.

03:56 So our final velocity will be zero because we'll reach the peak of our jump, and that means the velocity in the y direction will be zero.

04:03 And the initial velocity will be this velocity we got over here, 2 .59, plus negative 9 .8 times time.

04:17 We're looking for time, so let's do this.

04:21 Negative 2 .59 equals negative 9 .8.

04:27 So time is equal to 0 .264 seconds.

04:42 That's just the time for when he's going upwards in the air.

04:44 So we need the time for when he's going downwards.

04:48 And that's actually pretty easy.

04:53 And so we're just going to multiply that number by two because the time it takes for him to go up will be the same as the time it takes for him to go back down.

05:07 So if we multiply that number by two, our real time, will stay time total is equal to 0 .528 seconds and that's the time he'll be in the air...

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