A-SSE The Physics Professor


Alignments to Content Standards: HSA-SSE.A.1.b A-SSE.A.1

Task

A physics professor says: "Of course, it is easy to see that $$ L_0 \sqrt{1 - \frac{v^2}{c^2}} = 0 $$ when $v = c$."

  1. Give a possible explanation in terms of the structure of the expression on the left why the professor might say that.
  2. Assuming that $L_0$ and $c$ are positive, what is the greatest possible value of the expression on the left? Explain your answer in terms of the structure of the expression.

IM Commentary

The purpose of this task is to provide students practice in drawing conclusions about expressions they might encounter in classes outside mathematics, by parsing them in terms of their algebraic structure. Teachers might stress the subtle difference between "seeing" why the expression must be zero and the more mechanical process of algebraically simplifying the expression upon substituting $v=c$. Although part (b) might initially be though of as an optimization problem in a calculus course, some elementary reasoning with the structure of the expression gives the same answer in a more fluid and conceptual fashion.

For reference, this formula is for the length contraction of an object travelling near the speed of light, denoted $c$. While the constant $c$ in the problem is indeed a positive constant, its positivity is irrelevant to the task at hand since only its square, $c^2$, appears in the formula.

Solution

  1. When $v=c$, the fraction $\frac{v^2}{c^2}$ is 1 and so $\left(1-\frac{v^2}{c^2}\right)$ is 0. Since the square root of $0$ is $0$, the entire expression is $0$ when $v=c.$
  2. The greatest possible value of the expression is $L_0$. To see this, we observe that the term $1-\frac{v^2}{c^2}$ is always at most 1, since the quantity $\frac{v^2}{c^2}$ is always at least zero. Now the square root of a quantity which is at most 1 is again at most 1, and so when multiplied by $L_0$, we find that the entire expression is at most $L_0$. Note this value does indeed occur, when (and only when) $v=0$.

mprice says:

about 1 month

There is a meaningful discussion to be had when students see the statement "v=c". In a computer science class, this would be read "v is assigned the value of c", so the logical replacement would be to take all instances of v in the equation and replace them with c. Of course, mathematically either substitution accomplishes the goal (and the solutions posted here don't specify either strategy, just that the ratio must be 1). I can definitely imagine a student asking which value "should" be replaced, though.

Cam says:

29 days

Good point. I think from a mathematical point of view, it's preferable to have equality be symmetric, so that $v=c$ and $c=v$ are equivalent expressions (which, I agree, differs from the computer science interpretation). Under this interpretation, I'd encourage students not to think there is a "should" involved at all -- there often several algebraic deductions for the same fact, and while some might be more direct than others (not that this is the case here), that doesn't mean any are wrong.

Ashley Williams says:

about 2 years

Is it possible to have a solution posted?

Noah Heller says:

about 2 years

Is it important to say v<c or v=c, or at least make mention real solutions?

Kristin says:

about 2 years

The task statement does say that v=c. Did you have something else in mind?

Ellen Whitesides says:

over 2 years

Why does this task not have a published solution?