To determine the correct equation for acceleration from a velocity vs. time graph, we need to understand the basic concept of acceleration. Acceleration ([tex]\(a\)[/tex]) is defined as the change in velocity ([tex]\(\Delta v\)[/tex]) over the change in time ([tex]\(\Delta t\)[/tex]):
[tex]\[ a = \frac{\Delta v}{\Delta t} \][/tex]
Now, let's carefully examine each given option to see which one represents this equation:
1. [tex]\( a = \frac{t}{\Delta v} \)[/tex]
This equation suggests that acceleration is the total time divided by the change in velocity. However, this is incorrect because it inverts the relationship between time and velocity. Acceleration should be the change in velocity divided by the change in time, not the other way around.
2. [tex]\( m = \frac{y_2 - v_1}{x_2 - x_1} \)[/tex]
This equation looks like a slope calculation from coordinate geometry, but the variables [tex]\(y_2\)[/tex], [tex]\(v_1\)[/tex], [tex]\(x_2\)[/tex], and [tex]\(x_1\)[/tex] don't clearly define an acceleration relationship. Specifically, acceleration is not the slope between these arbitrary points on a graph unless they specifically represent velocity and time changes, which is not indicated.
3. [tex]\( a = \frac{\Delta v}{m} \)[/tex]
Here, [tex]\(a\)[/tex] is the acceleration, [tex]\(\Delta v\)[/tex] is the change in velocity, but [tex]\(m\)[/tex] is introduced, which does not clearly define any quantity related to time. Since [tex]\(m\)[/tex] does not represent a change in time, this equation is also incorrect.
4. [tex]\( m = \frac{x_2 - x_1}{y_2 - y_1} \)[/tex]
Similar to option 2, this equation describes a generic slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]. This equation does not represent the relationship of acceleration, as it does not specify velocity and time.
Given the examination of all options, the equation that appropriately matches the definition of acceleration ([tex]\(a = \frac{\Delta v}{\Delta t}\)[/tex]) is:
[tex]\[ \boxed{3} \][/tex]
This is the correct choice that most likely represents the determination of acceleration from a velocity vs. time graph.
