(II) A roller coaster reaches the top of the steepest hill...

(II) A roller coaster reaches the top of the steepest hill with a speed of 6.0 $\mathrm{km} / \mathrm{h}$ . It then descends the hill, which is at an average angle of $45^{\circ}$ and is 45.0 $\mathrm{m}$ long. Estimate its speed when it reaches the bottom. Assume $\mu_{\mathrm{k}}=0.18$ .

Key Concepts

Conservation of Energy

This concept involves equating the initial energy of a system with its final energy, taking into account all forms of energy transformation. In dynamics problems like a roller coaster descending a hill, the loss in gravitational potential energy is partly converted into kinetic energy, while some of it is dissipated due to non-conservative forces such as friction.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. When friction is present, the work done by friction (a non-conservative force) must be subtracted from the energy available for converting potential energy into kinetic energy.

Gravitational Potential Energy

Gravitational potential energy represents the energy stored due to an object’s position in a gravitational field. In the context of an inclined plane, the component of the weight of the object along the direction of the descent determines how much potential energy is converted to kinetic energy.

Kinetic Friction

Kinetic friction is the resistive force encountered by an object as it slides over a surface and is characterized by the coefficient of kinetic friction. This force performs negative work on the object, reducing the total mechanical energy and thus affecting the final speed of the object.

Transcript

00:01 We got a roller coaster here.

00:03 It doesn't look much like a roller coaster, but from a physics point of view, we can just make it a block.

00:10 And it reaches the top of a steep hill with a speed of six kilometers per hour.

00:17 It then descends the hill, which is at a angle of 45 degrees.

00:22 So more like this.

00:25 It descends, it goes over a distance that descent is 45 meters.

00:30 And the coefficient of friction between the, well, basically, it's not really between the, it's rolling resistance.

00:39 It's basically saying we have some rolling resistance here, which obviously there always is, but it's not really a coefficient of, what did they say? let's see here.

00:51 Yeah, it's not really, it's not, i mean, the wheels on a roller coaster don't slide along the track.

00:57 So, i mean, this is kind of a, it's a model for rolling resistance.

01:03 We're going to call it that it's probably not even a very good model because it probably has very little to do with the weight of the of the roller coaster you know depending if there's more people or not in it it probably doesn't change this much at all but so it's not probably a good model but we'll go with it so the friction force which is resisting the motion is mukk times n and is the weight times cost sine of theta, which is a force balance in this direction.

01:38 Force balance in this direction tells us that negative of the friction force plus the weight, sine of theta, equals the mass times acceleration in that direction.

01:48 Plugging this into here, this into here, and saying that using the weight is mg, we cancels out, and we get that the acceleration is g times sine of theta minus muke, cosine of theta.

02:03 Because the result, i think, we've seen in previous problems.

02:07 Plugging in the specific values we have, we get that the acceleration is 9 .682 meters per second squared...

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