course 201
9/11, 11:11amvvvv
I apologize for this being so late, I just had other assignments do with closer due dates.
1. State the definition of rate of change.
-The average rate of change is the change of A over the change of B.
2. State the definition of velocity.
-Average rate of change of position with respect to clock time.
3. State the definition of acceleration.
-The average rate of change of velocity with respect to clock time.
4. A ball rolls along a path, moving from position 20 cm to position 50 cm as its velocity increases from 5 cm/s to 15 cm/s.
• What is its change in velocity and how do you obtain it from the given information?
-The change in velocity is +10cm/s. Subtracting initial velocity from final velocity will give you the change in velocity.
• What is its change in position and how do you obtain it from the given information?
-The change in position is +30cm. Subtracting initial position from final position will give you the change in position.
5. A ball accelerates from velocity 30 cm/s to velocity 80 cm/s during a time interval lasting 10 seconds.
Explain in detail how to use the definitions you gave above to reason out
• the average velocity of the ball during this interval, -First off, you start out by adding Vf by Vo or final velocity by initial velocity. Vf-Vi = 80 cm/s + 30 cm/s = 110 cm/s. The next step is to divide 110cm/s by two, which will average the two speeds. (110 cm/s)/2 = 55cm/s. Now you have the answer of 55cm/s, which is the average velocity.
and
• its acceleration during this interval. –First off, you start out by subtracting Vf by Vo. 80cm/s – 30cm/s = 50cm/s. This gives you +50cm/s. Next, we divide the change of velocity by the time interval of this change, which is 10 seconds. 50cm/s / 10s = 5cm. Giving us the answer of the average rate of acceleration, which is +5cm/s.
Good, except that the units of your answer are wrong.
Remember, the main goal is to use a detailed reasoning process which connects the given information to the two requested results. You should use units with every quantity that has units, units should be included at every step of the calculation, and the algebraic details of the units calculations should be explained.
6. A ‘graph trapezoid’ has ‘graph altitudes’ of 40 cm/s and 10 cm/s, and its base is 6 seconds. Explain in detail how to find each of the following:
• The rise of the graph trapezoid. –To get the rise, all you do is subtract the lower alt. of the trapezoid by the higher alt. 40cm/s – 10cm/s gives us a rise of 30cm/s.
• The run of the graph trapezoid. –The run of the trapezoid is the length of the base, which is 6 seconds.
• The slope associated with the trapezoid. –The slope of the trapezoid is calculated by the rise divided by the run. So when we take the rise, 30cm/s, and divide it by the run, 6s, we get the slope of 5cm/s.
&#The units of your final result do not follow from the units of the quantities you used to calculate it. &#
• The dimensions of the equal-area rectangle associated with the trapezoid. –To find the dimensions of a rectangle, you must find the average alt. of the trapezoid, which is found by adding the two alt. together and dividing by two, 10cm/s + 40cm/s = 50cm/s / 2 = 25cm/s. Giving the rectangle the dimensions of 25cm/s by 6s.
• The area of the trapezoid. –To get the area, you must turn it into a rectangle like the problem above. 10cm/s + 40cm/s = 50cm/s / 2 = 25cm/s. Then multiple 25cm/s by 6s giving you 150cm/s^2, or the area of the trapezoid.
When you multiply cm/s by s you don't get cm/s^2.
Each calculation should include the units at every step, and the algebraic details of the units calculations should be explained.
7. If the altitudes of a ‘graph trapezoid’ represent the initial and final positions of a ball rolling down an incline, in meters, and the base of the trapezoid represents the time interval between these positions in seconds, then
• What is the rise of the graph trapezoid and what are its units? -The rise will be the product of subtracting Vo from Vf. The units will be meters.
• What is the run of the graph trapezoid and what are its unit? -The run is simply the length of the base, which will be measured in seconds.
• What is the slope of the trapezoid and what are its units? -The slope is found by finding the rise and dividing it by the run, the units will be m/s.
• What is the area of the trapezoid and what are its units? -You find the area by making it a rectangle (finding the average alt. of the trapezoid and then multiplying it against the run.) The units will be m/s^2.
&#There is an error in your units. &#
• What, if anything, does the slope represent? -It represents average rate of acceleration.
• What is the altitude of the equal-area rectangle and what are its units? -The alt. of the rectangle will be the average velocity of the trapezoid (Vf+Vo/2). It’s unit is m.
&#There is an error in your units. &#
• What is the base of the equal-area rectangle and what are its units? The base will stay the same as the trapezoid, it’s unit is seconds.
• What, if anything, does the area represent? Total displacement of the ball.
Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.
8. If the altitudes of a ‘graph trapezoid’ represent the initial and final velocities of a ball rolling down an incline, in meters / second, and the base of the trapezoid represents the time interval between these velocities in seconds, then
• What is the slope of the trapezoid and what are its units? -The slope is calculated by finding the rise (subtracting Vi from Vf) and dividing it by the run (length of the base.) It’s unit is m/s^2.
• What is the area of the trapezoid and what are its units? -The area is given by multiplying average alt. (adding Vf and Vo, then dividing by two) by the base. The unit is m/s^2.
• What, if anything, does the slope represent? -The slope represents average acceleration.
• What, if anything, does the area represent? -The area represents total displacement of the ball.
Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.
9. A ball rolls along a path, moving from position 20 cm to position 50 cm as its velocity increases from 5 cm/s to 15 cm/s.
• If its acceleration is uniform, then how long does this take, and what is the ball’s acceleration? Before we do anything else, we need to find out much time it took for the ball to move from 20 cm to 50 cm. To do that, we need to find the average velocity (5cm/s + 15cm/s / 2 = 10cm/s.) If the average velocity is 10cm/s, and the ball is displaced by a total of 30 cm (50cm – 20cm = 30cm of displacement), that means that it takes a total of 3 seconds for the ball to be displaced 30cm. Now we can solve for acceleration. First, you need to find the average rate of change in velocity (Vf-Vi = 15cm/s – 5cm/s = 10cm/s), then you divide the average rate of change in velocity by the change in clock time, which is 3 seconds. 10cm/s / 3s = 3.33cm/s^2. The average rate of acceleration for the ball is 3.33cm/s^2.
Very good. You did the right things, but near then end you didn't use the right words to describe them. Be sure you have the terminology right. vf - vi is the change in velocity, not an average rate of change; and you get acceleration by dividing change in velocity (not average rate of change in velocity) by change in clock time.
Overall you did very well, except on some of your labeling of quantities and your units calculations.
You could probably use a little more work on problems 7 and 8. I'm including an expanded version of these last two problems. The original questions are still marked with &&&& at the end. The questions marked with ## break the question into smaller questions, and it is these questions you should answer. You should insert your answers to the questions ending in ## into a copy of these 'expanded' problems and submit them.
7. If the altitudes of a ‘graph trapezoid’ represent the initial and final positions of a ball rolling down an incline, in meters, and the base of the trapezoid represents the time interval between these positions in seconds, then
What quantity is represented by the rise of the graph trapezoid and what are its units? &&&&
What quantities are represented by the altitudes of the trapezoid?
What are the units of the altitudes? ##
What quantity is represented by the base of this trapezoid, what are its units and what does it mean? ##
How is the rise of the trapezoid calculated, what are the meanings of the quantities used to calculate it, and what are the units of the rise? ##
What therefore is the meaning of the rise? ##
What quantity is represented by the run of the graph trapezoid and what are its units? &&&&
What quantity is represented by the slope of the trapezoid and what are its units? &&&&
This question should be answered base on the meanings and the units of the rise and the run. ##
How are the units of the slope calculated from the given information? ##
What quantity is represented by the area of the trapezoid and what are its units? &&&&
What quantity is represented by the altitude of the equal-area rectangle and what are its units? ##
What quantity is represented by the base of the equal-area rectangle and what are its units? ##
How are the altitude and base of the equal-area rectangle used to calculate the area of the trapezoid? ##
How are the units of the area calculated from the given information? ##
Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.
8. If the altitudes of a ‘graph trapezoid’ represent the initial and final velocities of a ball rolling down an incline, in meters / second, and the base of the trapezoid represents the time interval between these velocities in seconds, then
What quantity is represented by the rise of the graph trapezoid and what are its units? &&&&
What quantities are represented by the altitudes of the trapezoid? ##
What are the units of the altitudes? ##
What quantity is represented by the base of this trapezoid, what are its units and what does it mean? ##
How is the rise of the trapezoid calculated, what are the meanings of the quantities used to calculate it, and what are the units of the rise? ##
What therefore is the meaning of the rise? ##
What quantity is represented by the run of the graph trapezoid and what are its units? &&&&
What quantity is represented by the slope of the trapezoid and what are its units? &&&&
This question should be answered based on the meanings and the units of the rise and the run. ##
How are the units of the slope calculated from the given information? ##
What quantity is represented by the area of the trapezoid and what are its units? &&&&
What quantity is represented by the altitude of the equal-area rectangle and what are its units? ##
What quantity is represented by the base of the equal-area rectangle and what are its units? ##
How are the altitude and base of the equal-area rectangle used to calculate the area of the trapezoid? ##
How are the units of the area calculated from the given information? ##
Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.