Phy 231
Your 'cq_1_8.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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A ball is tossed upward with an initial velocity of 25 meters / second. Assume that the acceleration of gravity is 10 m/s^2 downward.
What will be the velocity of the ball after one second?
answer/question/discussion:
Answer: Gravity pushes down on the ball 10m/s/s. So in one second the balls velocity would be 10m/s less. The answer is 15m/s.
What will be its velocity at the end of two seconds?
answer/question/discussion:
Answer: The same principle as above question, 5m/s.
During the first two seconds, what therefore is its average velocity?
answer/question/discussion:
Answer: ave vel = (25m/s + 15m/s + 5m/s)/3 = 15m/s.
Since acceleration is uniform you would simply used the initial and final velocities on the interval, though in this case that would give you the same result as the one you obtained here.
In general if you use intermediate values as well as initial and final values in calculating an approximate mean, the 'endpoint' values are not equally weighted with the 'internal' values. Review the trapezoidal rule from calculus (where initial and final values only get half the weight of the intermediate values), and the idea that the average value of a function on an interval is equal to its integral on an interval divided by the length of the interval.
For a straight-line graph (e.g., for the linear v vs. t graph of uniform acceleration) the trapezoidal approximation is identical with the average of any number of equally-spaced values spanning the interval, and with the exact integral, so has previously mentioned in this particular instance where acceleration is uniform (implying a linear v vs. t graph), there is no error in using any of these methods.
How far does it therefore rise in the first two seconds?
answer/question/discussion:
Answer: distance ball has risen = 15m/s * 2s = 30m.
What will be its velocity at the end of a additional second, and at the end of one more additional second?
answer/question/discussion:
Answer: The velocity would 5m/s going downward and then another second would show a velocity of 15m/s going downward.
Velocities have directions. For motion in one dimension (e.g., as in this case up-down) this is indicated by + and - values, relative to the direction you have chosen as positive. Neither up or down can be chosen as the positive direction, but once chosen all vector quantities have to be expressed according to your choice. Relevant vector quantities include displacement, velocity and acceleration (as well as force, momentum, and a number of other directional quantities that are directly related to displacement, velocity and acceleration).
At what instant does the ball reach its maximum height, and how high has it risen by that instant?
answer/question/discussion:
Answer: It would be at a maximum height 2.5sec after the ball was released. Ave vel through 2.5s = (25m/s+0m/s)/2 = 12.5m/s. The distance bell has risen: 12.5m/s * 2.5s = 31.25m.
What is its average velocity for the first four seconds, and how high is it at the end of the fourth second?
answer/question/discussion:
Answer: ave vel through 4 sec: (25m/s + 15m/s + 5m/s + 5m/s) / 4 = 12.5m/s. The ball is 45m higher than when it was released.
The final velocity would be -5 m/s, and this would change your result.
How high will it be at the end of the sixth second?
answer/question/discussion:
Answer: The ball would be back at the initial point of release.
The ball would have passed the initial point of release, provided acceleration remains uniform. If the ball is initially released from level ground, then of course the ground gets in the way and this will be the case. However note that it was not assumed that the ball was released from ground level, so the possibility that it maintains its acceleration for the entire 6 seconds must be addressed.
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20 minutes
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I've inserted some notes related to the calculus of average values and to the trapezoidal rule. Let me know if you have questions on those concepts.
See also my other notes and please make any necessary revisions.
&#Please see my notes and submit a copy of this document with revisions and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end). &#