Which equation is most likely used to determine the acceleration from a velocity vs. time graph?
[tex]a=\frac{t}{\Delta v}[/tex]
[tex]m=\frac{v_2-v_1}{x_2-x_1}[/tex]
[tex]a=\frac{\Delta v}{m}[/tex]
[tex]m=\frac{x_2-x_1}{y_2-y_1}[/tex]
Answer (2)
Acceleration is the rate of change of velocity with respect to time, expressed as a = Δ t Δ v .
The slope of a velocity vs. time graph represents the acceleration.
The equation m = x 2 − x 1 v 2 − v 1 calculates the slope, where v 2 − v 1 is the change in velocity and x 2 − x 1 is the change in time.
Therefore, the most likely equation to determine acceleration from a velocity vs. time graph is m = x 2 − x 1 v 2 − v 1 .
Explanation
Problem Analysis We are asked to identify the equation that is most likely used to determine acceleration from a velocity vs. time graph.
Understanding Acceleration Recall that acceleration is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as: a = Δ t Δ v where:
a represents acceleration,
Δ v represents the change in velocity,
Δ t represents the change in time.
Analyzing the Options Now, let's examine the given options:
a = Δ v t : This equation is incorrect because it represents the inverse of acceleration.
m = x 2 − x 1 v 2 − v 1 : This equation represents the slope of a line on a graph. In the context of a velocity vs. time graph, where the y-axis represents velocity and the x-axis represents time, this equation becomes m = Δ t Δ v , which is the definition of acceleration.
a = m Δ v : This equation could be correct if m represents the change in time, Δ t . However, without knowing what m represents, it's less direct than option 2.
m = y 2 − y 1 x 2 − x 1 : This equation represents the inverse of the slope, so it is not correct for determining acceleration.
Conclusion Based on the analysis, the equation that is most likely used to determine acceleration from a velocity vs. time graph is: m = x 2 − x 1 v 2 − v 1 This equation directly calculates the slope of the velocity vs. time graph, which represents acceleration.
Examples
Imagine you are tracking the motion of a car on a straight road. By plotting the car's velocity at different points in time on a velocity vs. time graph, you can use the equation a = Δ t Δ v to determine the car's acceleration. This is crucial for understanding how the car's speed is changing and predicting its future motion, which is essential in fields like automotive engineering and traffic management.
The most likely equation used to determine acceleration from a velocity vs. time graph is m = t 2 − t 1 v 2 − v 1 . This equation reflects the definition of acceleration as the slope of the graph, representing the change in velocity over the change in time. Thus, it effectively calculates the acceleration of an object in motion.
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