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# The figure on the left above shows a wheel with a mark on its rim. The wheel i...

### Question

The figure on the left above shows a wheel with a mark on its rim. The wheel is rolling on the ground at a constant rate along a level straight path from a starting point to an ending point. The graph of y = d(t) on the right could represent which of the following as a function of time from when the wheel began to roll?

### Options

A)
The speed at which the wheel is rolling
B)
The distance of the wheel from its starting point
C)
The distance of the mark on the rim from the center of the wheel
D)
The distance of the mark on the rim from the ground

The correct answer is D.

### Explanation:

Choice D is correct. The graph on the right shows the change in distance from the ground of the mark on the rim over time. The y-intercept of the graph corresponds to the mark’s position at the start of the motion (t = 0); at this moment, the mark is at its highest point from the ground. As the wheel rolls, the mark approaches the ground, its distance from the ground decreasing until it reaches 0—the point where it touches the ground. After that, the mark moves up and away from the ground, its distance from the ground increasing until it reaches its maximum height from the ground. This is the moment when the wheel has completed a full rotation. The remaining part of the graph shows the distance of the mark from the ground during the second rotation of the wheel. Therefore, of the given choices, only choice D is in agreement with the given information.

Choice A is incorrect because the speed at which the wheel is rolling does not change over time, meaning the graph representing the speed would be a horizontal line. Choice B is incorrect because the distance of the wheel from its starting point to its ending point increases continuously; the graph shows a quantity that changes periodically over time, alternately decreasing and increasing. Choice C is incorrect because the distance of the mark from the center of the wheel is constant and equals the radius of the wheel. The graph representing this distance would be a horizontal line, not the curved line of the graph shown.