Question

The distance d that a certain particle moves may be calculated from the expression d = at + bt2 where a and b are constants; and t is the elapsed time. what must the dimensions of the quantities a and b be, respectively?

Answer 1

Guven that the distance d that a certain particle moves may be calculated from the expression $d = at + bt^2$ where a and b are constants; and t is the elapsed time.

Distance is a length and hence the dimension of distance is L.

Now, $at$ and $bt^2$ also will have the dimension of L.

Time has a dimension of T.

For $at$, let the dimension of $a$ be $X$, then
$XT=L \\ \\ \Rightarrow X=\frac{L}{T}$

For $bt^2$, let the dimension of $b$ be $Y$, then
$YT^2=L \\ \\ \Rightarrow Y= \frac{L}{T^2}$

Therefore, the dimension of $a$ is $\frac{L}{T}$ while the dimension of $b$ is $\frac{L}{T^2}$.