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=(L)/(T)`

For
`bt^2`, let the dimension of
`b` be
`Y`, then

`YT^2=L \\ \\ \Rightarrow Y= (L)/(T^2)`

Therefore, the dimension of
`a` is
`(L)/(T)` while the dimension of
`b` is
`(L)/(T^2)`.