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A Mathematical Model of Motion

Learn how to interpret, graph, and analyze motion using mathematical models. Discover how to draw graphs, write equations, and calculate velocity and displacement from position-time and velocity-time graphs.

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A Mathematical Model of Motion

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  1. A Mathematical Model of Motion

  2. Graphing Motion in One Dimension • Interpret graphs of position versus time for a moving object to determine the velocity of the object • Describe in words the information presented in graphs and draw graphs from descriptions of motion • Write equations that describe the position of an object moving at constant velocity

  3. Parts of a Graph • X-axis • Y-axis • All axes must be labeled with appropriate units, and values.

  4. Position vs. Time • The x-axis is always “time” • The y-axis is always “position” • The slope of the line indicates the velocity of the object. • Slope = (y2-y1)/(x2-x1) • d1-d0/t1-t0 • Δd/Δt

  5. Uniform Motion • Uniform motion is defined as equal displacements occurring during successive equal time periods (sometimes called constant velocity) • Straight lines on position-time graphs mean uniform motion.

  6. Given below is a diagram of a ball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds. Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

  7. The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

  8. Steepness of slope on Position-Time graph • Slope is related to velocity • Steep slope = higher velocity • Shallow slope = less velocity

  9. Different Position. Vs. Time graphs Constant positive velocity (zero acceleration) Constant negative velocity (zero acceleration)

  10. X B A t C A … Starts at home (origin) and goes forward slowly B … Not moving (position remains constant as time progresses) C … Turns around and goes in the other direction quickly, passing up home

  11. During which intervals was he traveling in a positive direction? During which intervals was he traveling in a negative direction? During which interval was he resting in a negative location? During which interval was he resting in a positive location? During which two intervals did he travel at the same speed? A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

  12. Graphing Velocity in One Dimension • Determine, from a graph of velocity versus time, the velocity of an object at a specific time • Interpret a v-t graph to find the time at which an object has a specific velocity • Calculate the displacement of an object from the area under a v-t graph

  13. Velocity vs. Time • X-axis is the “time” • Y-axis is the “velocity” • Horizontal lines = constant velocity

  14. Distance Traveled • Remember that Velocity = Δd Δt • Rearranging, we get Δd = velocity X Δt • So….the area underneath a velocity-time graph is equal to the displacement during that time period.

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