#$&*
course PHY 241
625pm 10/23/2011
Solving Uniform Acceleration ProblemsPossible 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:
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 # 6, 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 # 5, and solve it using the equations of uniformly accelerated motion.
What is the acceleration of a car at a uniform rate, that starts from rest travels 15 km in 5 minutes? What is the final velocity?
'ds = (v_f + v_0) / 2 * 'dt
'ds/'dt = (v_f + v_0) / 2
2('ds/'dt) - v_0 = v_f
v_f = 2(15 km / 5 min) - 0
= 6 km/min
a_ave = 'dv/dt
= (v_f - v_0) / 5 min
= (6km/min)/ 5 min
= 1.2 km/min^2
Then solve the same problem using the equations of uniformly accelerated motion.
`ds = v0 `dt + .5 a `dt^2
15 km = 0*(5min) + .5*a*(25 min^2)
a = 15 km/12.5 min^2
= 1.2 km/min^2
vf = v0 + a * `dt
= 0 + 1.2 km/min^2 * 5 min
= 6 km
Make up a problem for situation # 5, and solve it using the equations of uniformly accelerated motion.
An object starts from rest and accelerates at a uniform rate of 2 cm/s. What is the final velocity and total time it takes the ball to move 10 cm?
`ds = v0 `dt + .5 a `dt^2
10 cm = 0*(dt) + .5(2cm/s)*'dt^2
'dt = sqrt(10) s
= 3.16 s
vf^2 = v0^2 + 2 a `ds.
(v_f)^2 = 0 + 2*(2cm/s)*10cm
v_f = sqrt(40) cm/s
= 2sqrt(10)
= 6.32 cm/s"
@& You used the equations of motion for both of these. Be sure you know the difference between reasoning (which just uses the defintiions of velocity and acceleration, and works for most but not all cases where you know three of the quantities) and use of the equations (which work in all cases and are necessary in two of them).*@