Ambiguity in forumlas for average speed & average velocity

[e-zero]

I'm going to start with an example:

On her way to school, a child discovered that her loonie is missing and there is a hole in her pocket. She turned back and walked 24 m east along the sidewalk. Then, she stopped for 18 s and decided to head back to school. After walking 11 m west, she found the loonie. If the child walked at a speed of 0.25 m/s, calculate,

a) her average velocity while searching for the coin?
b) her average speed while searching for the coin?

Obviously there are two parts where the student travelled.

First part:
x1=24m
v1=0.25m/s
t1=96s (after calculation)

Second part:
x2=11m
v2=0.25m/s
t2=44s (after calculation)

If I wanted to find the average velocity then I would use my velocity formula: (avg)v = (x2 - x1) / t2 - t1

but I have to be careful because writing (avg)v = (11m - 24m) / (96s - 44s) is wrong!

Is this a common mistake? Should I name my variables differently?

 

[mikeph]

Why is the average velocity wrong? take the total displacement and divide by the total time

(24 - 11)/(96 + 18 + 44)

Do you think the answer is wrong because you didn't include the time she was stopped for?

 

[jbriggs444]

As I read the problem, the 18 seconds standing still should also be averaged in, but that's irrelevant to the question that I think you are asking.

The formula that you are using intends that t1 and t2 are the start time and the end time of an interval. The "t1 - t2" clause will then give the total duration of that interval.

The t1 and t2 that you have calculated are the durations of two sub-intervals instead. The total duration of the interval would be given by "t1 + t2" in your case.

Yes, you need to keep track of the quantities that your variables denote and whether those are the same quantities that your formulas need as inputs or produce as outputs. If careful choice of variable names makes it easier to keep track, then by all means, choose your variable names carefully. Or actually write down what you are using them to mean. Careful documentation earns points and is a good habit to cultivate.

There are concepts of weighted averages in general and time-weighted averages in particular that might be of use if you are interested in understanding this more deeply.

 

[e-zero]

So in this case I could just leave the variables as is?

 

[e-zero]

Specifically, should I rename my variables for x1, x2 and t1, t2?? Because if you were to plug them 'directly' into the formula you would get the wrong answer.

 

[mikeph]

It doesn't matter what you call them, as long as you understand the concept you will find the right answer. My advice is, never rely on an equation to be used blindly. In that equation, t1 and t2 are clock times, not time differences given in the question, so you need a new equation.

 

[e-zero]

Ok, but I come from a programming background if if that was a sequence if code, then that formula would spit out the wrong answer. I'm just going to name the variables differently, in this case, to avoid confusion or loss of marks.