In the equation v = at² - bt, we have two terms involving coefficients a and b. Let's determine the dimensions and SI units for these quantities:
Dimension and SI unit for 'a': In the equation, v = at² - bt, the term at² represents acceleration multiplied by time squared. The SI unit for acceleration is meters per second squared (m/s²), and the SI unit for time squared is seconds squared (s²). Therefore, the dimension of 'a' would be [acceleration / time squared], which is [m/s² / s²]. Simplifying this, we get [1 / s], which represents the dimension of frequency. Hence, 'a' has the dimension of frequency, and its SI unit is hertz (Hz).
Dimension and SI unit for 'b': In the equation, v = at² - bt, the term -bt represents a force multiplied by time. The SI unit for force is newton (N), and the SI unit for time is seconds (s). Therefore, the dimension of 'b' would be [force / time], which is [N / s]. Simplifying this, we get [kg · m / s² / s], which represents the dimension of mass divided by time cubed. Hence, 'b' has the dimension of [mass / time³], and its SI unit is kilogram per cubic meter per second (kg/m³/s).
To summarize:
- The dimension of 'a' is [1 / s], and its SI unit is hertz (Hz).
- The dimension of 'b' is [mass / time³], and its SI unit is kilogram per cubic meter per second (kg/m³/s).