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A suitable approach to this question involves analyzing the velocity-time graph in order to develop a verbal description of the moving object it represents. Then match the verbal description to the particular features of a position-time graph would be consistent with such a motion.
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The velocity-time graph displays a straight diagonal line located in the positive region of the graph. The line starts at the v=0 m/s level and slopes towards high velocity values. The slope of the line represents the acceleration of the object; the fact that this line is sloped means the object is changing its velocity. The object starts at rest (v=0 m/s) and accelerates to a high speed. The velocity is positive the entire time. Thus, the velocity-time graph represents an object that starts at rest and speeds up while moving in the positive direction.
A positive velocity is represented on a position-time graph by a line that slopes upward. An accelerated motion is represented on a position-time graph by a curved line. The slope of the line on a position-time graph represents the objects velocity (speed and direction). Since the object starts slow and finishes fast, the curve of the line on the position-time graph should start with zero slope and gradually become a steeper slope.
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Velocity-Time Graphs:
Velocity versus time graphs represent changes that occur in an object's velocity with respect to time. These graphs most commonly display horizontal lines or straight diagonal lines. Horizontal lines represent an object with a constant velocity. If the horizontal line is on the time axis, the object is at rest (a constant velocity of 0 m/s). If the horizontal line is above or below the time axis, then the object is moving with a constant positive or a constant negative velocity. Straight diagonal lines represent an accelerated motion. The slope of the line is the acceleration of the object.
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Position-Time Graphs:
Position versus time graphs represent changes that occur in an object's position with respect to time. These graphs most commonly display horizontal lines, straight diagonal lines and curved lines. Horizontal lines represent an object at rest. Straight diagonal lines represent a constant speed motion. Curved lines represent an accelerated motion. The slope of the line is the velocity of the object. Steeply-sloped lines represent fast-moving objects. Gently-sloped lines represent slow-moving objects.
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