In the following equations, the distance x is in metres, the time t in seconds and the velocity v in `(metres)/(second)`. What are the SI units of the constants `C_1` and `C_2` ?
(a). `v^2=2C_1x`
(b). `x=C_1cosC_2t`
(c ). `v=C_1e^(-C_2t)`

Explanation

(a) `(m)/(s)^2`, (b) `m`,`/(sec)`, (c ) `(m)/(s)`,`/sec`
(a) we have the expression `v^2=2C_1x`. The unit is left side and right side of any expression should be same.
unit of left side expression `=[(m)/(s)]=(m^2)/(s^2)`, hence unit of right side expression should also be same. `c_1xxm=(m^2)/(s^2)`, hence unit of `C_1=(m^2)/(s^2)`
(b) we have the expression `x=C_1cosC_2t` unit of left side expression `=m` on right side of expression `cosC_2t` is unitless. Hence unit of `C_1` should also be m. The expression `C_2t` should be unitless hence unit of `C_2` should be `per sec` of `/sec`.
(c ) we have the expression `v=C_1e^(-c_(2)t)`
The term `e^(-C_(2)t)` is unitless. The power term `C_(2)t` should also be unit less, hence the unit of `C_2` should be `per sec` or `/sec` the unit of `C_1` should be same as unit on left side of expression. Hence the unit of `C_1` should be`(m)/(s)`