course Phy231
2/24/10 7pm
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.
Problem #10
Let's say that an object has an acceralation of 3cm/s^2, it traveled 50cm in 10sec. So a=3cm/s^2, 'dt=50cm, 'ds=10sec
Therefore, 50cm/10s= 5cm/s is the vAve
Assuming the initial velocity is 0cm/s then I can find vf. (v0+vf)/2=vAve (0cm/s + Vf)/2=5cm/s (0m/s + vf)=10cm/s vf=10cm/s-0cm/s vf=10cm/s
I'm not sure where the acceleration fits into this because I just picked one.
You can't assume an initial velocity. v0 is not one of the three specified variables.
Pair up your known quantities and see what can be reasoned out.
Can you use a and `dt to find any of the seven quantities v0, vf, a, `ds, `dt, `dv and vAve?
What about a and `ds?
What about `dt and `ds?
One of these combinations doesn't tell you anything. The other two do.
Problem #8
vf=15cm/s a=0.5cm/s^2 'ds=75cm
(vf-v0)/'dt=a (15cm/s-v0)/'dt=0.5cm/s (15cm/s-v0)=0.5cm/s^2*'dt 15cm/s /0.5cm/s^2='dt+v0 30s='dt+v0 so if v0=0cm/s then 'dt=30s
I'm not sure how to find v0 if you didn't assume that v0=0cm/s"
You have to use the equations of uniformly accelerated motion to solve this problem.
List the four equations of uniformly accelerated motion.
Determine the equation(s) which include vf, a and `ds.
Each such equation will include an unknown variable, for which you should then solve.
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