Answer to Question #93529 in Mechanics | Relativity for Esmahan M. Muthana

Question

6. Curly, Moe, and Larry are 3 lab students tasked with finding the speed of a small, battery powered car. They decide to lay out a course for the car by putting 2 pieces of tape on the floor a known distance apart, timing how long between the time the car passes the first piece of tape and then the second tape. They will calculate speed by dividing the distance between the tapes by the time it took to make the trip. Curley votes to put the two pieces of tape 1 meter apart because it is easy and the distance really does not make a difference. He is outvoted by Moe and Larry, who vote for a distance of 5 meters, arguing that they make the same error in timing no matter the length of the course and dividing this error by a larger distance will give them a more accurate answer. With whom do you side? Defend your answer with a quantitative argument. You might want to estimate a timing uncertainty? Maybe assume some time?

Answer 1

Let the car passes $s$ meters per $t$ second with real speed $v$, where $s$ is a distance between 2 pieces of tape. Assume we have time error $\epsilon$, i.e. we can calculate speed of the car by formula

$$\tilde{v}=\frac{s}{t+\epsilon}.$$

If we increase the distance to $ns$, then the car needs $nt$ second and

$$\lim\limits_{n\to \infty} \frac{ns}{nt+\epsilon}=\frac{s}{t}=v.$$

Hence, the greater $n$ (and so the distance between 2 pieces of tape), the less error $|v-\tilde{v}|.$

So, we are on the side of Moe and Larry.