course Phy 121
1. State the definition of rate of change.vvvv
&&&&
The definition of rate of change (in terms of A with respect to B) is change in A over change in B, or:
Ave rate = delta A/delta B.
2. State the definition of velocity.
&&&&
Velocity is defined as the average rate of change of position (displacement) with respect to change in clock time (time interval).
3. State the definition of acceleration.
&&&&
It’s defined as the average rate of change of velocity with respect to change in 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? &&&&
It’s change in velocity is 10 cm/s, and I determined this by taking the difference of the two given velocities.
What is its change in position and how do you obtain it from the given information? &&&&
It’s change in position is 30cm, and I obtained this by finding the difference between the given positions.
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, &&&&
Since the velocities are already given, I assume that it’s the average of the two numbers (30 and 80), so it would be 55 cm/s, or did I misunderstand what you were asking for? If it’s just the difference then it would be 50cm/s
and
its acceleration during this interval. &&&&
Since acceleration is rate of change in in velocity over time interval, then in would look something like this:
(80cm/s - 30cm/s)/10 sec = 5 cm/s/s = 5 cm/s^2.
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.
Good. For clarification of your question on the very first part of the problem, and for general reference, see the note below:
`dg5
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. &&&&
The rise of the trapezoid is basically equal to the final velocity minus the initial velocity(the two altitudes), so if in this case, 10cm/s is initial and 40cm/s is final, then it would be 40cm/s - 10cm/s = 30cm/s.
The run of the graph trapezoid. &&&&
Run would be interpreted as the change in the clock time, so since the base is 6s, then the change would be 6s since the length of the entire base would represent change in time between the two altitudes (did I say that right?).
The slope associated with the trapezoid. &&&&
Slope is usually defined as rise over run, so: (40cm/s - 10cm/s)/6s = (30cm/s)/6s = 5cm/s^2 (acceleration!)
The dimensions of the equal-area rectangle associated with the trapezoid. &&&&
You can add the two altitudes or the trapezoid together and divide them by 2 in order to find the equivalent equal-area rectangle length, so it would be (40cm/s + 10cm/s)/2 = 25cm/s, so the dimensions of the equal area rectangle would be 25 cm/s by 6 s.
The area of the trapezoid. &&&&
This can be found using the same calculation as above, then multiplying it by delta t (length times width).
So area = 25cm/s * 6 s = 150 cm.
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 based 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 of the graph trapezoid would be found by taking the difference in the two altitudes, both in meters, then dividing it by the time interval in seconds, producing a value measured in meters per second (m/s).
What is the run of the graph trapezoid and what are its unit? &&&&
The run is basically the time interval between the initial and final velocities, so it would be equal to the length of the base measured in seconds.
What is the slope of the trapezoid and what are its units? &&&&
It’s going to be the change in the altitudes (velocities) measured in m/s, divided by the time interval in seconds, yielding a value measured in m/s divided by seconds, or m/s^2.
What is the area of the trapezoid and what are its units? &&&&
The area would be the initial velocity + the final velocity in m/s divided by 2 (no units), times the time interval (base) in seconds. The seconds in the equation “cancel out”, leaving a value measured in meters.
What, if anything, does the slope represent? &&&&
Seems to me that it represents acceleration.
What is the altitude of the equal-area rectangle and what are its units? &&&&
The altitude would be the initial and final velocities (in m/s) added together and divided by 2 (basically an average of the sides of the trapezoid). The 2 has no units, so the end result would still be in m/s.
What is the base of the equal-area rectangle and what are its units? &&&&
The base is the time interval measured in seconds.
What, if anything, does the area represent? &&&&
Not entirely sure, is it a representation of the distance traveled? Sounds good to me…
Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.
Just for clarification of a couple of minor points:
`dg7
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 based 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? &&&&
It’s still defined as rise over run, so it’d be the difference in the sides over the time interval, so it would be measured in m/s^2.
What is the area of the trapezoid and what are its units? &&&&
By finding the altitude of an equal area rectangle, you could find the area by multiplying said value in m/s times the base in seconds, yielding a value in meters. This is true because the definition of the area of a rectangle in length * width, and since it’s an ‘equal area’ rectangle, the area of the rectangle is equal to the area of the trapezoid.
What, if anything, does the slope represent? &&&&
Acceleration.
What, if anything, does the area represent? &&&&
Total distance.
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? &&&&
Hmm. I know that the answer to this problem is staring me right in the face, but for some reason I’m having trouble figuring it out. I’m going to kick myself when I see the answer. "
Average velocity is 10 cm/s, change in position is 30 cm so it takes 3 sec.
Change in velocity is 10 cm/s (coincidentally same as aveage velocity) so acceleration is 10 cm/s / (3s) = 3.33... cm/s^2.
*&$*&$