Open Question
Write a short description of the motion of a real object for which FIGURE EX1.20 would be a realistic position-versus-time graph.
2. 1D Motion / Kinematics
Conceptual Problems with Position-Time Graphs
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The position of an object vibrating on a moving spring is represented by the diagram below. Which of the following options is true of the velocity and acceleration at Point P?
A
The velocity is positive and the acceleration is positive
B
The velocity is positive and the acceleration is negative
C
The velocity is negative and the acceleration is positive
D
The velocity is negative and the acceleration is negative
Verified step by step guidance1
Identify the position of Point P on the graph. Point P is located at the bottom of the curve, indicating a minimum position on the x vs. t graph.
Determine the velocity at Point P. The velocity is the derivative of the position with respect to time, represented by the slope of the tangent to the curve at Point P. Since Point P is at a minimum, the slope of the tangent is zero, indicating that the velocity is zero at this point.
Consider the direction of motion just before and after Point P. As the object approaches Point P, it is moving downward, and after Point P, it starts moving upward. This indicates that the velocity changes from negative to positive, suggesting that the velocity is negative just before reaching Point P.
Analyze the acceleration at Point P. Acceleration is the derivative of velocity with respect to time, or the second derivative of position. At a minimum point on a position-time graph, the acceleration is positive because the curve is concave upwards, indicating that the object is accelerating upwards.
Conclude the analysis: At Point P, the velocity is negative just before reaching the minimum, and the acceleration is positive as the object is about to move upwards. Therefore, the correct statement is: 'The velocity is negative and the acceleration is positive.'
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