Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.
Kinematics is the study of motion. In this lesson, we will practice calculating the two types of velocity and acceleration. Also, we will be looking at graphs of displacement vs. time and acceleration vs. time.
Two Types of Velocities
When you are driving in your car, the speedometer tells you your instantaneous speed, which is the speed you are traveling at that instant in time. Most modern cars have the feature where your average speed is given, which is the total distance you have driven divided by the time it took to drive that distance. I am using the word 'speed' because the speedometer and the car's computer do not report the direction you are going along with the speeds. Velocity is a vector and requires speed and direction.
The first type of velocity we will look at is average velocity, or an object's displacement over a period of time. Determining average velocity only requires the use of the equation:
Here's an example:
Determine the average velocity of an object that traveled 30 meters north, 20 meters south, and then 15 meters east in 20 seconds.
First, we need to determine the displacement of the object, which requires the vector addition of the distances. Let's assign j to be a vector pointing north and i to be a vector pointing east.
Numerically determining the displacement, we get:
Note that we replaced 20 meters south with -20 meters north. This means our total displacement s is equal to 15 meters east plus 10 meters north.
Now we can plug the displacement and travel time into the average velocity equation:
Determining instantaneous velocity, or an object's velocity at a specific point in time, is a little more involved. If the position of an object varies with time, we can take the first derivative of the position-time function to determine the instantaneous velocity at any time.
There are a few steps to take a derivative:
1. Multiply the coefficient of the variable by the exponent on the variable, and write that number as the variable's new coefficient.
2. Lower the original exponent by 1, and write that as the new exponent.
C is the coefficient.
t is the time variable.
n is the exponent on the time variable.
3. Do steps 1 and 2 for each part of the equation that has the variable t in it. The derivative of constants is zero.
Let's do an example:
A mass in an experiment has the position function:
Determine its velocity at 6 seconds.
Taking the derivative of this function will give us the velocity function at any time. Following the derivative rules, we get
To solve the problem, all we have to do is plug in 6 seconds in for time:
Acceleration is the change in velocity divided by the time required for the change. To determine instantaneous acceleration, you would take the derivative of the velocity-time function and plug in any time. Let's do an example.
Let's do an example:
It takes an object 6 seconds to go from 396 m/s to 1656 m/s in the x-direction. What is the object's average acceleration?
Using the average acceleration equation, we get:
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We just finished calculating velocity and acceleration. Velocity is represented on a displacement versus time graph, but we have to calculate it. Slope on any graph is the change in y divided by the change in x.
Looking at the graph, we see the y-axis is displacement, and the x-axis is time.
Putting the y-units over the x-units we get m/s, which is velocity. This means that the slope of a displacement versus time graph is velocity. Let's determine the velocity of the object at 11.5 seconds.
The point on the graph corresponding to 11.5 seconds is part of a straight-line segment. Any point along that line-segment has the same slope.
We can calculate the slope of that line-segment using the slope formula, and the answer will be the instantaneous velocity of the object.
Acceleration vs. Time Graph
Acceleration versus time graphs tell us the acceleration at any time, the change in acceleration, and the change in velocity. Slope on this graph is the change in acceleration, which is known as jerk. The area between the graph and the x-axis is the change in velocity.
The area under the horizontal line at 2 m/s2 between 0 seconds and 4 seconds is a rectangle.
The area of a rectangle is length times width. Let's calculate the area of the shaded area:
Assuming the object started at rest, it will be moving at 8 m/s after 4 seconds.
Instantaneous speed is how fast something is moving at a specific instant in time.
Average speed is the total distance an object has moved divided by the total time it took to move that distance.
Average velocity is the displacement of an object divided by the time it took for the displacement to occur.
Velocity is a vector, and direction must be included with vectors.
Instantaneous velocity is the speed and the direction an object is moving at a particular instant in time. It can be calculated by taking the derivative of a position-time function, and then plugging in any time value.
Acceleration is the change in velocity over time.
Displacement versus time graphs tell us instantaneous velocity, which is the slope of the graph at the point in question.
The slope on acceleration versus time graphs is the change in acceleration, or jerk. The area between the graph and the x-axis is the change in velocity.
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