A student is skateboarding down a ramp that is 6.0 m long...

A student is skateboarding down a ramp that is 6.0 m long and inclined at $18^{\circ}$ with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is $2.6 \mathrm{m} / \mathrm{s} .$ Neglect friction and find the speed at the bottom of the ramp.

Key Concepts

Geometric Relationships on Inclined Planes

This involves using trigonometric relationships to relate the dimensions of the ramp to the vertical height change. Specifically, the sine of the incline angle multiplied by the ramp length gives the change in height, which is essential for calculating the gravitational potential energy change.

Kinetic Energy

Kinetic energy is the energy of motion, given by the formula one-half mass times the square of its velocity. In energy conservation problems, the increase in kinetic energy as an object moves down an incline is directly calculated from the loss of gravitational potential energy.

Conservation of Energy

This concept states that energy in a closed system remains constant over time, transforming between forms but never being lost. In frictionless scenarios, like the problem described, gravitational potential energy converts entirely into kinetic energy, allowing the calculation of final speed by equating the sum of initial kinetic energy and potential energy decrease to the final kinetic energy.

Gravitational Potential Energy

This is the energy an object possesses due to its position relative to a reference point, typically calculated as the product of mass, gravitational acceleration, and height. In inclined plane problems, the height can be related to the length of the ramp and its angle of inclination, making it a crucial component in determining the energy available for conversion into kinetic energy.

Transcript

00:01 This problem, we have a skateboard going down the ramp.

00:04 That's 6 meters long, so call that l.

00:09 The skateboard initially has a velocity of 2 .6 meters per second, and we want to calculate the final velocity once it reaches the bottom of the ramp.

00:23 So to find the final velocity, we can use a kinematic equation, which is the final velocity squared equals so initial velocity squared plus two times the acceleration times the distance, or in this case, will be the length of the ramp.

00:43 So we have everything on the right except for the acceleration.

00:48 And we can find the acceleration by newton's second law, which is the sum of the forces acting on skateboard equals mass times acceleration.

00:58 So we need to find the forces acting on the skateboard, and in this case, will be the x -axis, because that's the direction of motion.

01:08 And then we can solve for the acceleration, and then plug it back into our equation, a kinematic equation, which we'll call equation 1.

01:18 So we need to find first a free body diagram on the skateboard, which again is going down some ramp.

01:28 So it has the weight of the skateboard going straight down.

01:34 And that means because the skateboard is at an angle, the weight has components in the x and y direction as well.

01:41 So we have the y direction, wy, and then to the right, wx.

01:48 And again, all we care about is the forces in the x direction.

01:52 And so all we have here is just wx.

01:57 So when we do the sum of the forces, we simply have wx.

02:02 Wx equals mass times acceleration...

Please give Ace some feedback