In the equation [tex]\( z = ct + d \)[/tex], [tex]\( z \)[/tex] is measured in meters and [tex]\( t \)[/tex] is measured in seconds. What are the dimensions (units) of [tex]\( c \)[/tex]?

A. [tex]\( \frac{m}{s} \)[/tex]

B. [tex]\( s \)[/tex]

C. [tex]\( \frac{s}{m} \)[/tex]

D. [tex]\( m \cdot s \)[/tex]

E. [tex]\( m \)[/tex]



Answer :

To determine the units of [tex]\( c \)[/tex] in the equation [tex]\( z = c \cdot t + d \)[/tex], we need to carefully analyze the units involved in each term of the equation.

1. Given Units:
- [tex]\( z \)[/tex] is measured in meters (m).
- [tex]\( t \)[/tex] is measured in seconds (s).
- [tex]\( d \)[/tex] is a constant and, like [tex]\( z \)[/tex], is measured in meters (m).

2. Isolate [tex]\( c \)[/tex]:
We can rearrange the equation [tex]\( z = c \cdot t + d \)[/tex] to better understand the units of [tex]\( c \)[/tex].

[tex]\[ z - d = c \cdot t \][/tex]

Since [tex]\( z \)[/tex] and [tex]\( d \)[/tex] are both in meters, [tex]\( z - d \)[/tex] will also be in meters (m).

3. Solve for [tex]\( c \)[/tex]:

[tex]\[ c = \frac{z - d}{t} \][/tex]

4. Substitute Units:
- The numerator [tex]\( z - d \)[/tex] has the units of meters (m).
- The denominator [tex]\( t \)[/tex] has the units of seconds (s).

[tex]\[ c = \frac{\text{meters}}{\text{seconds}} = \frac{m}{s} \][/tex]

Therefore, the units of [tex]\( c \)[/tex] are meters per second (m/s). [tex]\(\boxed{\frac{m}{s}}\)[/tex]