A laboratory cart moves on a straight horizontal track with an initial speed of away…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Torque and Rotational Dynamics · Torque and Rotational Dynamics

Problem

A laboratory cart moves on a straight horizontal track with an initial speed of

1.5\,\text{m}\,\text{s}^{-1}

away from a motion detector. Simultaneously, a solid disk mounted on a low-friction axle begins rotating from rest with a constant angular acceleration of

4.0\,\text{rad}\,\text{s}^{-2}

. The physics instructor provides a verbal description of a graph that will be projected but is not shown to the students: • The cart’s velocity–time graph is a horizontal line at

+1.5\,\text{m}\,\text{s}^{-1}

for

0\le t\le1.0\,\text{s}

, then a straight line decreasing uniformly to

0\,\text{m}\,\text{s}^{-1}

t=4.0\,\text{s}

. • The disk’s angular-velocity–time graph is a straight line through the origin with slope

4.0\,\text{rad}\,\text{s}^{-2}

for the entire

4.0\,\text{s}

interval. Using the instructor’s description as the stimulus, answer the following questions in the order given. Show all reasoning, including how areas and slopes are interpreted. (a) Construct a qualitative sketch (no numerical scales required) of the angular-displacement–time graph for the disk over the

4.0\,\text{s}

interval. Describe the key features of your graph in words. (b) Calculate the disk’s total angular displacement after

4.0\,\text{s}

. (c) Determine the cart’s total linear displacement during the same time interval. (d) Explain, making explicit reference to the shapes and areas of the two graphs, why the mathematical forms of the linear and angular kinematic equations ensure consistent results when mapping from one motion to the other.

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