Question

1. (5pts) you are given an equation for time, t in terms of distance, r, speed, v, and a constant, k. $t^{2}=\frac{12k r^{2}}{v^{3}}$. what are the units of the constant k in terms of meters and seconds?

Explanation:

Step1: Identify the dimensions of each variable

Time $t$ has dimension of seconds (s), distance $r$ has dimension of meters (m), and speed $v$ has dimension of $\frac{m}{s}$.

Step2: Rewrite the given equation for dimensions

The given equation is $t^{2}=\frac{42k r^{2}}{v^{3}}$. In terms of dimensions, substituting $[t]=s$, $[r]=m$ and $[v]=\frac{m}{s}$, we get $(s)^{2}=\frac{k\times(m)^{2}}{(\frac{m}{s})^{3}}$.

Step3: Solve for the dimension of $k$

First, simplify the right - hand side: $\frac{k\times(m)^{2}}{(\frac{m}{s})^{3}}=\frac{k\times m^{2}}{\frac{m^{3}}{s^{3}}}=k\times\frac{s^{3}}{m}$. Then, from $(s)^{2}=k\times\frac{s^{3}}{m}$, we can solve for $k$ by cross - multiplying. We get $k = \frac{m}{s}$.

Answer:

The units of $k$ are $\frac{m}{s}$