To determine the acceleration from a velocity versus time graph, we need to use the fundamental definition of acceleration. Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, it is expressed as:
[tex]\[ a = \frac{\Delta v}{\Delta t} \][/tex]
Where:
- [tex]\( \Delta v \)[/tex] is the change in velocity.
- [tex]\( \Delta t \)[/tex] is the change in time.
Now let's evaluate each given equation against this definition:
1. [tex]\[ a = \frac{t}{\Delta v} \][/tex]
- This equation implies acceleration is the time divided by the change in velocity, which is not consistent with the correct definition of acceleration.
2. [tex]\[ m = \frac{y_2-v_1}{x_2-x_1} \][/tex]
- This equation seems to resemble a form of the slope of a line but does not directly correspond to the relationship between velocity and time specific to acceleration.
3. [tex]\[ a = \frac{\Delta v}{m} \][/tex]
- This equation suggests that acceleration is the change in velocity divided by some undefined quantity [tex]\( m \)[/tex]. Without clarity on what [tex]\( m \)[/tex] represents, this cannot be correct.
4. [tex]\[ m = \frac{x_2-x_1}{y_1-y_1} \][/tex]
- This equation is mathematically incorrect as it implies [tex]\( \frac{0}{0} \)[/tex] since [tex]\( y_1 - y_1 = 0 \)[/tex].
Given the above analysis, none of the options appear to perfectly match the traditional definition of acceleration. However, considering the context and the closest match to the correct form [tex]\( a = \frac{\Delta v}{\Delta t} \)[/tex], we infer:
[tex]\[ \textbf{a} = \frac{\Delta v}{t} \][/tex]
Accordingly, the correct choice, even if not perfect, seems to be:
[tex]\[ \boxed{1} \][/tex]
