To determine the acceleration from a velocity vs. time graph, we need to recall the basic definition of acceleration in physics. Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as:
[tex]\[ a = \frac{\Delta v}{\Delta t} \][/tex]
where:
- [tex]\( \Delta v \)[/tex] represents the change in velocity
- [tex]\( \Delta t \)[/tex] represents the change in time
Given the provided options:
1. [tex]\( a = \frac{t}{\Delta v} \)[/tex]
2. [tex]\( m = \frac{v_2 - v_1}{x_2 - x_1} \)[/tex]
3. [tex]\( a = \frac{\Delta v}{m} \)[/tex]
4. [tex]\( m = \frac{x_2 - x_1}{y_2 - y_1} \)[/tex]
Let’s analyze each option:
1. [tex]\( a = \frac{t}{\Delta v} \)[/tex]: This equation suggests that acceleration [tex]\( a \)[/tex] is the time [tex]\( t \)[/tex] divided by the change in velocity [tex]\( \Delta v \)[/tex]. This is incorrect because it inverts the relationship between acceleration, time, and velocity change.
2. [tex]\( m = \frac{v_2 - v_1}{x_2 - x_1} \)[/tex]: This equation resembles the formula for the gradient of a line in coordinate geometry, where [tex]\( m \)[/tex] represents the slope. Here, [tex]\( v_2 \)[/tex] and [tex]\( v_1 \)[/tex] are velocities, and [tex]\( x_2 \)[/tex] and [tex]\( x_1 \)[/tex] are positions. This does not appropriately address acceleration from a velocity vs. time graph.
3. [tex]\( a = \frac{\Delta v}{m} \)[/tex]: This equation is not in the standard form for acceleration. 'm' is typically used to denote mass, slope, or another term, but it’s not used in this context to derive acceleration.
4. [tex]\( m = \frac{x_2 - x_1}{y_2 - y_1} \)[/tex]: This formula also describes the slope of a line in coordinate geometry, but it uses different variables that pertain to position and another coordinate. It doesn’t relate to acceleration directly either.
To find the correct formula, we should remember that acceleration is mathematically defined by the change in velocity divided by the change in time. Considering this, none of the provided equations directly represent the standard definition of acceleration accurately. However, if we re-examine option 3 with better syntax:
[tex]\[ a = \frac{\Delta v}{t} \][/tex]
Assuming a typographical adjustment for clarity, it becomes evident that this modified version demonstrates the correct relationship:
Thus, option 3 with the clarification matches the form closest to:
[tex]\[ a = \frac{\Delta v}{\Delta t} \][/tex]
Therefore, the most accurate equation for determining acceleration from a velocity vs. time graph is:
[tex]\[ a = \frac{\Delta v}{t} \][/tex]
From the provided options, it is clear that the correct answer would indeed match the reinterpreted option more clearly. Based on this analysis, the correct option determining acceleration is the reinterpreted answer:
Option 3 [tex]\( a = \frac{\Delta v}{\Delta t} \)[/tex], showing 3 is the number associated with the right choice.
