Free response · Unit 1 Beta

A cart's round trip on a ramp

Translation Between Representations · 12 points

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The question

Name: ______________________   Date: ____________   A cart's round trip on a ramp (12 points)

Diagram

A cart pushed up a ramp

A cart at the bottom of a long, straight ramp is given a quick push and leaves the bottom (x = 0) moving up the ramp with speed v₀. It rolls up, stops for an instant, and rolls back down to the bottom. Throughout the trip after the push, the cart's acceleration has a constant magnitude a and points down the ramp. Up the ramp is the positive x direction.

v₀ (up the ramp)bottom of the ramp (x = 0)+x

Key: gold arrow, the cart's velocity just after the push.

  1. (a) Draw a motion diagram for the cart, from just after the push until it returns to the bottom: dots for the cart's position at equal time intervals, with a velocity arrow at each dot. Draw the trip up and the trip down side by side so they do not overlap, mark the turning point, and draw one arrow showing the direction of the cart's acceleration. [3 points]

  2. (b) Derive an expression for the time T the cart takes to return to the bottom of the ramp, in terms of v₀ and a. Begin your derivation by writing a fundamental equation, and make sure the signs are consistent with your motion diagram. [3 points]

  3. (c) Sketch a graph of the cart's velocity as a function of time from just after the push until it returns to the bottom. Label v₀ and the time T, and mark where the cart is at the top of the ramp. [3 points]

  4. (d) Explain how your motion diagram, your expression for T and your graph are consistent with one another. Refer to a specific feature of each representation. [3 points]