course Phy 121
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Solving Uniform Acceleration Problems
Possible Combinations of Variables Direct Reasoning
Using Equations Problem
Possible Combinations of Variables
There are ten possible combinations of three of the the five variables v0, vf, a, Dt and Ds. These ten combinations are summarized in the table below:
If we know the three variables we can easily solve for the other two, using either direct reasoning or the equations of uniformly accelerated motion (the definitions of average velocity and acceleration, and the two equations derived from these by eliminating Dt and then eliminating vf).
Only two of these situations require equations for their solution; the rest can be solved by direct reasoning using the seven quantities v0, vf, a, Dt, Ds, Dv and vAve. These two situations, numbers 5 and 8 on the table, are indicated by the asterisks in the last column.
Direct Reasoning
We learn more physics by reasoning directly than by using equations. In direct reasoning we think about the meaning of each calculation and visualize each calculation.
When reasoning directly using v0, vf, `dv, vAve, `ds, `dt and a we use two known variables at a time to determine the value of an unknown variable, which then becomes known. Each step should be accompanied by visualization of the meaning of the calculation and by thinking of the meaning of the calculation. A 'flow diagram' is helpful here.
Using Equations
When using equations, we need to find the equation that contains the three known variables.
* We solve that equation for the remaining, unknown, variable in that equation.
* We obtain the value of the unknown variable by plugging in the values of the three known variables and simplifying.
* At this point we know the values of four of the five variables.
* Then any equation containing the fifth variable can be solved for this variable, and the values of the remaining variables plugged in to obtain the value of this final variable.
Problem
Do the following:
* Make up a problem for situation # 3, and solve it using direct reasoning.
* Accompany your solution with an explanation of the meaning of each step and with a flow diagram.
* Then solve the same problem using the equations of uniformly accelerated motion.
* Make up a problem for situation # 8, and solve it using the equations of uniformly accelerated motion.
Ans:
* Given v0, vf, and `ds find a and `dt.
To find the acceleration use the change in velocity and divide it by the change in time. Use the following equation: a = (vf - v0) / (t2 - t1)
To find the change in time, simply subtract the two given times.
'dt = t2 - t1
You can't find `dt unless you are given t1 and t2.
You are given only v0, vf and `ds.
How do you find `dt given only that information?
* flow diagram for this problem:
From vf and v0 we get `dv, shown by lines from vf and v0 at the top level to `dv. From vf and v0 we also get and vAve, shown by similar lines running from v0 and vf to vAve. From a, there are two lines that connect to 'dv and 'dt. Then from vAve and `dt we get `ds, with the accompanying lines indicating from vAve and `dt to `ds. Finally, from `dv and `dt we get acceleration.
Good description of the flow diagram, but you say that 'from vAve and `dt we get `ds'; this is not helpful because you don't know `dt at this point. But the given information does allow you to get vAve, and you are given `ds, so you can use vAve and `ds to get `dt.
* Using the 4 equations of uniform acceleration
Used this equation to solve for 'dt: vf = v0 + a * 'dt
Once I rearrange the equation using algebra I come up with a new equation that looks like this: (vf - v0) / a = 'dt.
You can't use this equation because you are not given a or `dt.
Next, I Used the same equation to find acceleration. Again, I had to rearrange the equation to look like this: (vf - v0) / 'dt = a.
* Given vf a and `ds find v0 and `dt using the 4 equations of uniform acceleration.
To find initial velocity (v0) use the equation vf = v0 + a * 'dt.
Given vf, a and `ds, you can't use this equation. You don't know v0 or `dt.
It must be rearranged to look like: vf - a * 'dt = v0. To find the change in time (`dt) I useed the same equation and rearranged it as follows: (vf - v0) / a = 'dt"
You gave good descriptions of the process you would use, but the steps you describe won't work for the given information.
Please give this another try and send me your analysis.
It's great to do this with symbols, but you should make up some actual numerical quantities, complete with units, and make sure your symbolic analysis works with the numerical quantities.