Acceleration. 3. Changing Direction occurs when the velocity and acceleration are neither parallel nor anti-parallel
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1 Acceleration When the velocity of an object changes, we say that the object is accelerating. This acceleration can take one of three forms: 1. Speeding Up occurs when the object s velocity and acceleration are in the same direction (parallel) 2. Slowing Down occurs when the object s velocity and acceleration are in opposite directions (antiparallel) 3. Changing Direction occurs when the velocity and acceleration are neither parallel nor anti-parallel We define acceleration as the rate of change of velocity. It can be calculated by taking the change in velocity and dividing by time: a = Δv Δt or a = v f v i t Example 1 A person is running. The runner changes velocity, going from +5.0 m / s to +7.0 m / s in a time of 1.0 s. What is the runner s acceleration?
2 Velocity-Time Graphs Assume the runner in Example 1 starts at rest ( v i = 0.0 m / s) at a time of t = 0.0 s. Given the acceleration that we calculated above, the following data table describes the motion of the runner over a 5.0 s interval: Time s ( ) Velocity ( m / s) This data can be plotted on a graph of velocity versus time Velocity (m/s) Time (s) As was the case with a position-time graph, there is a great deal of information that can be determined from a velocity-time graph.
3 Some of the things you can determine directly from a graph of velocity versus time include: describing the motion of the object determining the velocity of the object at a given time o by reading the graph determine the time when the object was moving with a given velocity o by reading the graph Some of the things you can determine indirectly that is, by doing some calculations from a velocity-time graph include: determine the distance traveled for an interval of time o area between the graph and the x-axis determine the displacement for an interval of time o area between the graph and the x-axis (area above axis is positive, below axis is negative) determine the speed of the object over a given time interval (average speed) o total distance (area) divided by time determining the velocity of the object over a given time interval (average velocity) o total displacement (area) divided by time determining the acceleration of the object at a given instant (instantaneous acceleration) o slope of the graph at that instant o if the line is curved, use a tangent line determining the acceleration of the object over a given time interval (average acceleration) o slope of a line joining the point on the graph at the start of the interval to the point on the graph at the end of the interval
4 Example 2 Use the velocity-time graph below to answer the questions. Note: Right is the positive direction Velocity (m/s) Time (s) 1. Describe the motion of the object.
5 2. What is the velocity of the object at t = 30 s? 3. At what time is the object s velocity 6 m / s [ Left ]? 4. Determine the distance traveled by the object for the entire trip t = 0 s to t = 50 s. 5. Determine the average speed of the object for the entire trip. 6. Determine the displacement of the object for the entire trip.
6 7. Determine the average velocity of the object for the entire trip. 8. Determine the acceleration of the object at t = 15 s and at t = 45 s. 9. Determine the average acceleration of the object for the entire trip.
7 Velocity-Time Graphs The following graph describes the motion of a car Velocity (m/s) Time (s) 1. Complete the data table below showing time intervals and position. Time ( s) Position ( m) Complete the data table below showing time intervals and acceleration. Time ( s) 0.0 to to to 12.0 Acceleration ( m / s 2 )
8 The following velocity-time graph shows the possible motion of a bicycle B 10 A C 5 Velocity (m/s) D F E -20 Time (s) 3. Complete the table showing time and position. Assume the bicycle starts at 0.0 m and 0.0 s. Section Time ( s) Displacement ( m) Position at the End of the Interval A B C D E F
9 4. Complete the table showing time and acceleration. Section A B C D E F Time ( s) Change in Velocity ( m / s) Acceleration ( m / s 2 )
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