(Note: If you are having trouble with this exercise, see Exercises for the Perplexed at the end of this document).

For each situation state which of the five quantities v0, vf, `ds, `dt and a are given, and give the value of each. 

• A ball accelerates uniformly from 10 cm/s to 20 cm/s while traveling 45 cm.

Here you know v0, vf and `ds, with respective values 10 cm/s, 20 cm/s and 45 cm.

• From v0 and vf, since accel is uniform you can easily find vAve.

• Then you can use the definition of vAve along with the value of `ds to obtain `dt.

• Finally you can determine `dv from v0 and vf and use the definition of acceleration, along with `dt, to find a.

• A ball accelerates uniformly at 10 cm/s^2 for 3 seconds, and at the end of this interval is moving at 50 cm/s.

Here you know a, `dt and vf, with respective values 10 cm/s^2, 3 sec and 50 cm/s.

• From a and `dt you can find `dv.

• From vf and `dv you can find v0.

• From v0 and vf you can find vAve, which with `dt allows you to find `ds.

• A ball travels 30 cm along an incline, starting from rest, while accelerating at 20 cm/s^2. 

Here you know v0, a and `ds, with respective values 0 (i.e., starting from rest), 20 cm/s^2 and 30 cm.

You can't directly reason out anything from these quantities:

• To use the definition of vAve you would need to know either vf (to be used with v0) or `dt (to be used with `ds).

• To use the definition of aAve you would have to know either vf (to be used with v0 to find `dv, so that you have a and `dv) or `dt (with a and `dt you can find `dv, then from v0 can find vf).

You can use the third or fourth equation of motion so get a result.

• However the intuitive meaning of the third equation (`ds= v0 `dt + 1/2 a `dt^2) and the fourth (vf^2 = v0^2 + 2 a `ds) are not intuitively clear and directly related to the definitions of velocity and acceleration.

• So while you would get a good solution, it would not have been obtained using direct reasoning in terms of the definition.  There's nothing wrong with the solution, it just isn't directly reasoned.

Then for each situation answer the following:

• Is it possible from this information to directly determine vAve? 

In the first situation you can directly determine vAve from the given quantities v0 and vf.

In the second situation you can't determine vAve directly from the given information, but in a couple of steps you can directly reason out all the quantities. 

In the third situation you can't get anywhere with direct reasoning, so you have to use the third and/or fourth equations of motion.

Is it possible to directly determine `dv? 

In the first situation you can directly determine `dv from the given quantities v0 and vf.

In the second situation you can directly determine `dv from a and `dt.

In the third situation you can't get anywhere with direct reasoning, so you have to use the third and/or fourth equations of motion.