PTR Example - Calculate and Justify

Prediction:

Justification:

a) The speeds of the red buggy (R) and the blue buggy (B) were determined by measuring the time interval for the each buggy to move a displacement of 2m. Speed = ∆x / ∆t for each buggy. Because both buggies can be modeled as constant velocity, I built an algebraic model of the form x = vt + xinitial. Because the blue buggy is pointed in the negative direction, the value of vB is negative, but I don't actually substitute this negative value until Step c, for reasons I'll justify later.

b) The buggies collide when the two variables x and t have the same value in the algebraic models for both buggies, therefore I set the equations equal to each other, in order to solve for t. My goal here was to solve the problem for a general case, so that it would be easy to consider the uncertainty of each buggy's velocity in Step c.

c) Through multiple time measurements, I was able to estimate the uncertainty of the speed of each buggy as ± 0.01m/s. Therefore, I can assume that the difference in these speeds has an uncertainty of ± 0.02m/s. I used this uncertainty to determine the maximum and minimum value for t of the buggy collision, resulting in a range of about 0.5s.

d) To solve for the collision position, I can use this range of t values to substitute back into the algebraic model for the red buggy, resulting in a range of possible values for position. I see that there is a range of about 0.2m in my prediction.

Limitations, Tradeoffs, Alternate Explanations (Justification continued)

The uncertainty in each buggy's speed is significant, partly because neither buggy travels exactly straight. We are able to use a foot to steer the buggies in a straighter path, but this probably slows the buggy down. Also, the battery power to the buggies probably isn't consistent, so the fact that we did our measurements for the prediction a few days before the test might be a factor. Ideally, we'd have a more consistent power source for the buggy, but having to account for uncertainty is actually a positive thing.

I realize that I did not account for the uncertainty of the red buggy velocity when I substituted back into the algebraic model in Step d. While this is technically incorrect, I made the decision that the uncertainty in the predicted time was sufficient.

Interestingly, the "timeless" solution involves less computation, and less annoying keeping track of units. This solution involves some cool trickery to compare the speeds of the two buggies as a ratio instead of using the actual speeds themselves, BUT once you solve this part the solution is more straightforward to write up and justify.

Limitations, Tradeoffs, Alternate Explanations (Justification continued)

The uncertainty in each buggy's speed is significant, partly because neither buggy travels exactly straight. We are able to use a foot to steer the buggies in a straighter path, but this probably slows the buggy down. Also, the battery power to the buggies probably isn't consistent, so the fact that we did our measurements for the prediction a few days before the test might be a factor. Ideally, we'd have a more consistent power source for the buggy, but having to account for uncertainty is actually a positive thing.

I realize that I did not account for the uncertainty of the red buggy velocity when I substituted back into the algebraic model in Step d. While this is technically incorrect, I made the decision that the uncertainty in the predicted time was sufficient.

Interestingly, the "timeless" solution involves less computation, and less annoying keeping track of units. This solution involves some cool trickery to compare the speeds of the two buggies as a ratio instead of using the actual speeds themselves, BUT once you solve this part the solution is more straightforward to write up and justify.