Question

which equation is most likely used to determine the acceleration from a velocity vs. time graph?
○ ( a = \frac{t}{delta v} )
○ ( m = \frac{v_2 - v_1}{x_2 - x_1} )
○ ( a = \frac{delta v}{m} )
○ ( m = \frac{x_2 - x_1}{v_2 - v_1} )

Explanation:

Step1: Recall acceleration from velocity - time graph

Acceleration \(a\) is defined as the rate of change of velocity with respect to time. In a velocity - time (\(v - t\)) graph, the slope of the graph gives the acceleration. The slope \(m\) of a line between two points \((t_1,v_1)\) and \((t_2,v_2)\) (where \(x\) - axis is time \(t\) and \(y\) - axis is velocity \(v\)) is given by \(m=\frac{v_2 - v_1}{t_2 - t_1}\), and since acceleration \(a\) is the slope of \(v - t\) graph, we can also think in terms of the formula for slope. Let's analyze each option:

• Option 1: \(a=\frac{t}{\Delta v}\) is incorrect because acceleration is \(\frac{\Delta v}{\Delta t}\), not \(\frac{t}{\Delta v}\).
• Option 2: \(m = \frac{v_2 - v_1}{x_2 - x_1}\) is the slope of a graph where \(y\) - axis is velocity and \(x\) - axis is position, which is not relevant for velocity - time graph.
• Option 3: \(a=\frac{\Delta v}{m}\) is incorrect as it is not the formula for acceleration from a \(v - t\) graph (this looks like a mis - arranged form of \(F = ma\) but not related to \(v - t\) graph slope).
• Option 4: \(m=\frac{x_2 - x_1}{v_2 - v_1}\) is incorrect. Wait, no, wait. Wait, in a velocity - time graph, the \(x\) - axis is time \(t\) and \(y\) - axis is velocity \(v\). So the slope \(m\) (which is acceleration) is \(\frac{\Delta v}{\Delta t}\). But if we consider the formula for slope between two points \((t_1,v_1)\) and \((t_2,v_2)\), the slope \(m=\frac{v_2 - v_1}{t_2 - t_1}\). But looking at the options, the fourth option is \(m=\frac{x_2 - x_1}{v_2 - v_1}\)? No, wait, maybe there is a typo in the variable names. Wait, maybe the \(x\) here is \(t\) (time). If we assume that the \(x\) in the options is a typo for \(t\), then \(m=\frac{v_2 - v_1}{t_2 - t_1}\) (which is the slope, i.e., acceleration). But among the given options, the second option is \(m=\frac{v_2 - v_1}{x_2 - x_1}\) and the fourth is \(m=\frac{x_2 - x_1}{v_2 - v_1}\). Wait, no, let's re - express.

Wait, acceleration \(a=\frac{\Delta v}{\Delta t}\). In a velocity - time graph, the slope of the line (which is acceleration) is calculated as the change in velocity (\(\Delta v=v_2 - v_1\)) divided by the change in time (\(\Delta t=t_2 - t_1\)). The formula for the slope \(m\) of a line is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In a velocity - time graph, \(y\) - axis is velocity (\(v\)) and \(x\) - axis is time (\(t\)). So \(m=\frac{v_2 - v_1}{t_2 - t_1}\), which is equal to acceleration \(a\). Now, looking at the options, the second option is \(m=\frac{v_2 - v_1}{x_2 - x_1}\). If we consider that \(x\) here represents time (maybe a variable name mix - up), then this formula represents the slope of the velocity - time graph, which is acceleration. Wait, no, the fourth option is \(m=\frac{x_2 - x_1}{v_2 - v_1}\), which would be the reciprocal. Let's check the other options:

• First option: \(a=\frac{t}{\Delta v}\) is wrong as \(a=\frac{\Delta v}{\Delta t}\).
• Third option: \(a=\frac{\Delta v}{m}\) is wrong, it's not the formula for acceleration from \(v - t\) graph.
• Second option: \(m=\frac{v_2 - v_1}{x_2 - x_1}\) - if \(x\) is time, then this is the slope (acceleration).
• Fourth option: \(m=\frac{x_2 - x_1}{v_2 - v_1}\) - this is the reciprocal of the slope, so wrong.

Wait, maybe the variable \(x\) in the options is a typo for \(t\) (time). So the formula for the slope (which is acceleration) of a velocity - time graph is \(m=\frac{v_2 - v_1}{t_2 - t_1}\), which is the second option \(m=\frac{v_2 - v_1}{x_2 - x_1}\) (assuming \(x\) is \(t\)).

Answer:

\(m=\frac{v_2 - v_1}{x_2 - x_1}\) (the second option)