Question

2.5 dimensional analysis
consider the equation:
where s is a length and t is a time. what are the dimensions and si units of the following?
(a) s0, (b) v0, (c) a0, (d) j0, (e) s0, and (f) c?
s = s0 + v0t + \frac{a0t^2}{6}+\frac{s0t^4}{24}+\frac{ct^5}{120}

Explanation:

Step1: Recall dimensional analysis rules

In SI units, length has dimension $L$ and time has dimension $T$. For an equation to be dimensionally consistent, each term on the right - hand side must have the same dimension as the left - hand side.

Step2: Analyze $s_0$

Since $s$ is a length and $s_0$ is added directly to other terms equal to $s$, the dimension of $s_0$ is $L$ and its SI unit is $m$.

Step3: Analyze $v_0$

We know that $v_0t$ must have the dimension of length. Since $t$ has dimension $T$, if $[v_0t]=L$ and $[t] = T$, then the dimension of $v_0$ is $\frac{L}{T}$ and its SI unit is $\frac{m}{s}$.

Step4: Analyze $a_0$

We have $\frac{a_0t^2}{6}$ with dimension of length. If $[\frac{a_0t^2}{6}]=L$ and $[t]=T$, then $[a_0]=\frac{L}{T^2}$ and its SI unit is $\frac{m}{s^2}$.

Step5: Analyze $j_0$ (assuming $j_0$ is a mis - print and you mean the coefficient of $\frac{S_0t^4}{24}$)

For the term $\frac{S_0t^4}{24}$ to have dimension of length, if $[\frac{S_0t^4}{24}]=L$ and $[t]=T$, then the dimension of the coefficient of $t^4$ (assuming it's $j_0$) is $\frac{L}{T^4}$ and its SI unit is $\frac{m}{s^4}$.

Step6: Analyze $S_0$

Since the term $\frac{S_0t^4}{24}$ has dimension of length and $[t]=T$, the dimension of $S_0$ is $\frac{L}{T^4}$ and its SI unit is $\frac{m}{s^4}$.

Step7: Analyze $c$

For the term $\frac{ct^5}{120}$ to have dimension of length, if $[\frac{ct^5}{120}]=L$ and $[t]=T$, then the dimension of $c$ is $\frac{L}{T^5}$ and its SI unit is $\frac{m}{s^5}$.

Answer:

(a) Dimension: $L$, SI unit: $m$
(b) Dimension: $\frac{L}{T}$, SI unit: $\frac{m}{s}$
(c) Dimension: $\frac{L}{T^2}$, SI unit: $\frac{m}{s^2}$
(d) Dimension: $\frac{L}{T^4}$, SI unit: $\frac{m}{s^4}$
(e) Dimension: $\frac{L}{T^4}$, SI unit: $\frac{m}{s^4}$
(f) Dimension: $\frac{L}{T^5}$, SI unit: $\frac{m}{s^5}$