#$&*
course Phy 241
7/21/2013 11:30 pm
Possible Combinations of VariablesThere 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:
1
v0
vf
a
2
v0
vf
dt
3
v0
vf
ds
4
v0
a
dt
5
v0
a
ds
*
6
v0
dt
ds
7
vf
a
dt
8
vf
a
ds
*
9
vf
dt
ds
10
a
dt
ds
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 # 10, 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.
#10
a, `dt, `ds
v_0 = 5m/s
v_f = 20m/s
t_0 = 3s
t_f = 6s
`d v = v_f-v_0
`d t = t_f-t_0
a = `d v/ `d t
First the change in velocity is found
then the change in time is found
then the change in velocity is divided by the change in time to find acceleration
`d v = v_f-v_0 = 20m/s-5m/s = 15m/s
`d t = t_f-t_0 = 6s-3s = 3s
a = `d v/ `d t = 15m/s/3s = 5m/s^2
@&
For this situation you need to specify the values of a, `dt and `ds, which you then use to find the values of the remaining quantities.
You appear to have assumed v0, vf and `dt. You did reason out a correctly from that information, but you did not reason out `ds.
In any case you need to also solve the given problem.
*@
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a = 5m/s^2
`d t = 3s
`d s = 37.5
v_0 = ( `d s-0.5*a* `d t^2)/ `d t = [37.5-0.5*(5m/s^2)*3s^2]/3s = 5m/s
v_f = v_0+a `d t = 5m/s+5m/s^2*3s = 20m/s
First, initial velocity is found by using a form of the equation `ds = v0 `dt + .5 a `dt^2
Then the final velocity is found using the equation v_0+a `d t.
In a flow diagram, the first level is `d t, v_0, and v_f.
In the second level, `d v can be found using v_0 and v_f
In the third level, a can be found using `d v and `d t.
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@&
You used equations rather than direct reasoning.
Direct reasoning:
vAve = 37.5 m / (3 s) = 12.5 m/s
`dv = 3 s * 5 m/s^2 = 15 m/s.
Half of `dv is 7.5 m/s so
vf = 12.5 m/s + 7.5 m/s = 20 m/s
v0 = 12.5 m/s - 7.5 m/s = 5 m/s.
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#8
v_f, a, `ds
v_f = 15m/s
a = 5m/s^2
`ds = 3m
I cannot figure this one out. Help?
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List the equations of uniformly accelerated motion.
Identify which equation(s) contain(s) the three given quantities. If there are two, select one.
Solve the equation for the unknown variable.
Plug in your quantities and find the value of that variable.
Then use a different equation, or direct reasoning (as you choose; both options will be available to you) to find the value of the fifth variable.
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#8
v_f, a, `ds
v_f = 15m/s
a = 5m/s^2
`ds = 3m
v_f^2 = v_0^2+2*a* `d s
v_f^2-2*a* `d s = v_0^2
sqrt(v_f^2-2*a* `d s) = v_0
sqrt((15m/s)^2-2*(5m/s^2)*3m) = v_0
v_0 = sqrt((15m/s)^2-2*(5m/s^2)*3m) ≈ 13.964 m/s
v_f = v_0+a* `d t
(v_f-v_0)/a = `d t
`d t ≈ (15m/s-13.964 m/s)/5m/s^2 ≈ 1.036m/s/5m/s^2 = 0.2072s
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&#Good work. See my notes and let me know if you have questions. &#