Angela has taught college microbiology and anatomy & physiology, has a doctoral degree in microbiology, and has worked as a post-doctoral research scholar for Pittsburgh’s National Energy Technology Laboratory.
Kinematics with the X,Y Graph
The basic x, y graph has many interesting applications in physics and kinematics. In addition to succinctly describing the straight line motion of an object, you can use these graphs to calculate information like displacement, velocity, and acceleration. In this lesson, we'll continue our examination of the velocity vs. time graph and how it can be used to calculate the acceleration of an object in straight line motion.
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Acceleration = the Slope of the Graph
You should be familiar with the velocity vs. time graph. This example graph illustrates how the velocity of a car changes as it drives along a straight track.
The equation for acceleration is a = Δv / Δ t. Remember, when using an equation with a delta (Δ), you need to calculate the change: Δ = final value - initial value
Δv = final v - initial v
Δt = final t - initial t
But, velocity (v) is on the y axis and time (t) is on the x axis. So, we could also write this equation as:
a = Δy / Δ x
Does this look familiar? It should. This is the equation for the slope of a line on an x, y graph. So, the slope of a velocity vs. time graph gives the acceleration over that section of the graph. Confused? Let's look at an example.
Acceleration Example Problem
This velocity vs. time graph shows the motion of a rat running in a long, straight tube.
First, the equation for acceleration is:
a = Δv / Δ t
Let's fill in what we know:
Δv = 20 m/s - 0 m/s = 20 m/s. His velocity started at 0 m/s and ended at 20 m/s so the change in velocity (Δv) was 20 m/s.
Δ t = 4 s - 0 s = 4 s. The time started when he started moving (0 seconds) and we only care about the first 4 seconds, so his change in time (Δ t) was 4 s.
Filling in the equation, we get:
a = (20 m/s) / (4 s) = 5 m/s^2
His acceleration over the first 4 seconds was 5 m/s^2. Remember, acceleration is a vector quantity and needs a directional component. In the case of straight line motion, the vector direction is equal to the sign of the magnitude. Since the 5 m/s^2 acceleration is positive, the vector direction is forward.
If we look at the last four seconds of the motion (t = 8s - 12s), the math is going to be nearly identical, but the change in velocity will be slightly different.
Δv = 0 m/s - 20 m/s = - 20 m/s
If we substitute this into the equation, we get:
a = (-20 m/s) / (4 s) = -5 m/s^2
Notice that the sign is negative. This means the acceleration is negative and the rat is slowing down. The negative sign is our vector direction.
Let's briefly review.
You can use a velocity vs. time graph to calculate the acceleration of an object in straight line motion. The slope of a velocity vs. time graph gives the acceleration over a specific section of the graph. The equation for the slope of a line is slope = Δy / Δ x
But, our axes have specific designations: y = velocity and x = time. Putting these together, we get the equation for acceleration: a = Δv / Δ t
Remember, the sign of the acceleration represents the directional component required of vector quantities. A positive acceleration means the object is speeding up and a negative acceleration means it's slowing down.
Review this lesson to learn how to:
- Use a velocity vs. time graph to calculate acceleration
- Compare positive acceleration with negative acceleration
- Solve an example problem, keeping in mind that acceleration needs a directional component
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Determining Acceleration Using the Slope of a Velocity vs. Time Graph
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