Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-2| Given Positions, Find Velocity and Acceleration

Learning Goal: To understand how to graph position, velocity, and acceleration of an object starting with a table of positions vs. time.

Problem:

The table shows the x coordinate of a moving object. The position is tabulated at 1-s intervals. The x coordinate is indicated below each time. You should make the simplification that the acceleration of the object is bounded and contains no spikes.

time (s) 0 1 2 3 4 5 6 7 8 9
x (m) 0 1 4 9 16 24 32 40 46 48

PART A. Which of the graph below best represents the function x(t), describing the object’s position vs. time?

There are four graphs showing position as a function of time. The position on all graphs is from 0 to 50 meters on the vertical axis. On graph 1 and graph 4, time ranges from 0 to 10 seconds on the horizontal axis. On graph 2 and graph 3, time ranges from 0 to 9 seconds on the horizontal axis. On graph 1, the position equals zero meters from 0 seconds to approximately 2 seconds, then it increases smoothly to approximately 6 seconds and 25 meters forming a convex curve and then also increases to 10 seconds and 45 meters forming a concave curve. On graph 2, the position equals zero meters from 0 seconds to approximately 1 seconds, then it increases smoothly to approximately 5 seconds and 25 meters forming a convex curve and then also increases to 9 seconds and 45 meters forming a concave curve. Graph 3 consists of 10 separate points with the trend line same as for graph 2. Graph 4 consists of 11 separate points with the trend line same as for graph 1.

ANSWER: Graph 2

A bounded and nonspiky acceleration results in a smooth graph of x vs. t.

PART B. Which of the following graphs below best represents the function v(t), describing the object’s velocity as a function of time?

There are four graphs which show the velocity as a function of time. Time ranges from 0 to 9 seconds on the horizontal axis. The velocity is from -8 to 8 meters per second on the vertical axis. All graphs consist of three horizontal segments and several vertical segments, connecting ends of horizontal segments. The first segment on all the graphs connects the point (0; 0) and the point (4; 8). The second segment on graphs 1 and 2 connects the point (4; 0) and the point (7; 0), and on graphs 3 and 4 connects the point (4; 8) and the point (7; 8). The third segment on graphs 1 and 4 connects the point (7; 8) and the point (9; 0), on graphs 2 and 3 connects the point (7; -8) and the point (9; 0).

ANSWER: Graph 4

The velocity toward the end of motion is positive and decreasing. This narrows the options down to either graph 1 or 4.

Two of the possible velocity vs. time graphs indicate zero velocity between t=4 and t=7s. The implications of the zero velocity means that the position vs. time graph is a horizontal line.

The problem states that “the acceleration of the object is bounded and contains no spikes.” This means that the velocity has no discontinuities.

PART C. Which of the following graphs in the figure below best represents the function a(t), describing the acceleration of this object?

There are four graphs showing the acceleration of a moving object as a function of time. Time ranges from 0 to 9 seconds on the horizontal axis. The acceleration is from -4 to 4 meters per second squared on the vertical axis. All graphs consist of three horizontal segments and several vertical segments, connecting ends of horizontal segments. The first segment on graphs 1 and 3 connects the point (0; 2) and the point (4; 2), on graph 2 connects the point (0; -2) and the point (4; -2) and on graph 4 connects the point (0; 0) and the point (4; 4). The second segment on graphs 1, 2 and 3 connects the point (4; 0) and the point (7; 0) and on graph 4 connects the point (4; 4) and the point (7; 4). The third segment on graphs 1 and 2 connects the point (7; 4) and the point (9; 4), on graph 3 connects the point (7; -4) and the point (9; -4), on graph 4 connects the point (7; 4) and the point (9; 4).

ANSWER: Graph 3

Acceleration is the time derivative of velocity. Toward the end of the motion the acceleration is negative.

The acceleration over the interval during which the object travels at constant speed is 0 m/s².

Acceleration is the time derivative of velocity. Initially the acceleration is positive.

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