course Phy 201
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
v0vfa
2
v0vf dt
3
v0vf ds
4
v0 adt
5
v0 a ds*
6
v0 dtds
7
vfadt
8
vfa ds*
9
vf dtds
10
adtds
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#1 If a ball accelerates at a uniform rate of 5cm/s^2 and covers a distance of 60cm in 4secs then what is its v0 and vf?
we know that if we travel 60cm in 4secs we have an vAve of 15cm/s
we also know if we travel at 5cm/s^2 for 4secs we will have a `dv of 20cm/s
knowing that this is uniformly acc. motion we know that v0,vAve,and vf are equally spaced and we have `dv of 20cm/s we can split this and add it to both sides of the vAve. which would be v0=5cm/s and vf=25cm/s
v0=`ds/`dt-.5a`dt
v0=60m/4s-.5(5cm/s^2)`dt
v0=15cm/s-10cm/s
v0=5cm/s
vf=a*`dt+v0
vf=5cm/s^2*4secs+5cm/s
vf=20cm/s+5cm/s
vf=25cm/s
Problem#2 Determine v0 and Dt for an object which accelerates through a distance of 60cm, Ending at a velocity 25cm/s and accelerating at 5cm/s^2
v0^2=vf^2-2a`ds
v0^2=625-600
v0^2=25
sqrt25
v0=5cm/s
vAve=(vf+v0)/2
vAve=15cm/s
`dt=`ds/vAve
`dt=4secs
&#Very good responses. Let me know if you have questions. &#