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title: "A block is initially at rest on top of an inclined ramp that makes an angle \\( \\theta_0 \\) with the horizontal. The distance measured along the base of the ramp is \\( D \\). After the block is released from rest, it slides down the frictionless ramp and then continues onto a rough horizontal surface until it finally comes to rest at the position \\( x = 4D \\) measured from the base of the ramp. The coefficient of kinetic friction between the block and the rough horizontal surface is \\( \\mu_k \\).(a) On the axes provided, sketch and label graphs of the following quantities as a function of the position \\( x \\) of the block for \\( -D \\le x \\le 4D \\). Both graphs must use the same vertical scale. i. The kinetic energy \\( K \\) of the block ii. The gravitational potential energy \\( U_g \\) of the block–Earth system(b) The block is now released from the top of a different ramp that still makes the same angle \\( \\theta_0 \\) with the horizontal but whose base length is \\( 2D \\). A student is asked whether the block’s final horizontal position will now be twice as far (i.e., at \\( x = 8D \\)) compared with the original situation. The student reasons that, because the new height is twice the original height, the block will have more energy at the base of the new ramp and therefore will slide farther along the horizontal surface until stopping at \\( x = 8D \\). i. Which aspects of the student’s reasoning, if any, are correct? If none are correct, write “none”. ii. Which aspects of the student’s reasoning, if any, are incorrect? If none are incorrect, write “none”.(c) Derive an equation for the new final position of the block in terms of \\( D \\).(d) Referring to the mathematical relationships you obtained in part (c): • For any correct aspects identified in part (b)(i), explain how your relationships support the student’s reasoning. • For any incorrect aspects identified in part (b)(ii), explain how your relationships correct the student’s reasoning."
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url: "https://nerd-notes.com/ubq/110166/"
date_modified: "2026-03-30T07:35:08+00:00"
---

# A block is initially at rest on top of an inclined ramp that makes an angle \( \theta_0 \) with the horizontal. The distance measured along the base of the ramp is \( D \). After the block is released from rest, it slides down the frictionless ramp and then continues onto a rough horizontal surface until it finally comes to rest at the position \( x = 4D \) measured from the base of the ramp. The coefficient of kinetic friction between the block and the rough horizontal surface is \( \mu_k \).(a) On the axes provided, sketch and label graphs of the following quantities as a function of the position \( x \) of the block for \( -D \le x \le 4D \). Both graphs must use the same vertical scale. i. The kinetic energy \( K \) of the block ii. The gravitational potential energy \( U_g \) of the block–Earth system(b) The block is now released from the top of a different ramp that still makes the same angle \( \theta_0 \) with the horizontal but whose base length is \( 2D \). A student is asked whether the block’s final horizontal position will now be twice as far (i.e., at \( x = 8D \)) compared with the original situation. The student reasons that, because the new height is twice the original height, the block will have more energy at the base of the new ramp and therefore will slide farther along the horizontal surface until stopping at \( x = 8D \). i. Which aspects of the student’s reasoning, if any, are correct? If none are correct, write “none”. ii. Which aspects of the student’s reasoning, if any, are incorrect? If none are incorrect, write “none”.(c) Derive an equation for the new final position of the block in terms of \( D \).(d) Referring to the mathematical relationships you obtained in part (c): • For any correct aspects identified in part (b)(i), explain how your relationships support the student’s reasoning. • For any incorrect aspects identified in part (b)(ii), explain how your relationships correct the student’s reasoning.

A block is initially at rest on top of an inclined ramp that makes an angle \( \theta_0 \) with the horizontal. The distance measured along the base of the ramp is \( D \). After the block is released from rest, it slides down the frictionless ramp and then continues onto a rough horizontal surface until it finally comes to rest at the position \( x = 4D \) measured from the base of the ramp. The coefficient of kinetic friction between the block and the rough horizontal surface is \( \mu_k \).

![A right-triangle ramp is drawn on the left side of a horizontal baseline. The ramp’s hypotenuse slopes downward from left to right at an angle labeled “θ0” between the hypotenuse and the rightward horizontal baseline. The horizontal run of the ramp (its base) is marked as length D, starting at x = –D on the leftmost top point of the ramp and ending at x = 0 at the foot of the ramp. From x = 0 to x = 4D, the baseline is a straight horizontal line representing the rough surface. Tick marks on the baseline indicate positions x = 0, D, 2D, 3D, and 4D. The inclined segment is frictionless; the horizontal segment is shaded or marked as rough. No block is explicitly drawn on the diagram.](https://nerd-notes.com/wp-content/uploads/ubq-diagrams/nerd-notes-physical-setup-110166-1774645452-scaled.jpg)

**Part a)** i. Sketch and label a graph of the block’s kinetic energy \( K \) versus position \( x \) for \( -D \le x \le 4D \), using an appropriate common vertical scale.   ii. Sketch and label a graph of the gravitational potential energy \( U_g \) of the block–Earth system versus position \( x \) for \( -D \le x \le 4D \), using the same vertical scale as in the previous graph. *(3 points)*

**Part b)** The block is released from the top of a new ramp that has a base length of \(2D\), but still makes an angle \(\theta\) with the horizontal. A student is asked to predict whether the final horizontal position of the block will be twice as far from the base of the ramp compared to when it was released from the original ramp. The student reasons that since the block will be released from a new height that is twice as high as the original height, the block will have more energy when it reaches the base of the ramp, so it will slide farther along the right surface before stopping at a new position. 1. State the aspects of the student’s reasoning that are correct, or write “none” if no aspects are correct. 2. State the aspects of the student’s reasoning that are incorrect, or write “none” if no aspects are incorrect. *(3 points)*

**Part c)** Derive an expression for the block’s new final horizontal position after sliding from the taller ramp, expressing your answer in terms of \( D \). *(3 points)*

**Part d)** Explain how the mathematical relationships from part (e) justify any correct aspects you listed and correct any incorrect aspects you listed in parts (b)(i) and (b)(ii). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110166/*