We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity , usually called simply velocity. It is the average velocity between two points on the path in the limit that the time and therefore the displacement between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x t.
The expression for the average velocity between two points using this notation is. After inserting these expressions into the equation for the average velocity and taking the limit as. The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t :.
Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point. Figure shows how the average velocity. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times,. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.
The graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid. During the time interval between 0 s and 0. In the subsequent time interval, between 0. From 1. The object has reversed direction and has a negative velocity. In everyday language, most people use the terms speed and velocity interchangeably.
In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar. We can calculate the average speed by finding the total distance traveled divided by the elapsed time:. Average speed is not necessarily the same as the magnitude of the average velocity, which is found by dividing the magnitude of the total displacement by the elapsed time. For example, if a trip starts and ends at the same location, the total displacement is zero, and therefore the average velocity is zero.
The average speed, however, is not zero, because the total distance traveled is greater than zero. If we take a road trip of km and need to be at our destination at a certain time, then we would be interested in our average speed. However, we can calculate the instantaneous speed from the magnitude of the instantaneous velocity:.
Some typical speeds are shown in the following table. When calculating instantaneous velocity, we need to specify the explicit form of the position function x t. The following example illustrates the use of Figure. Calculate the average velocity between 1.
Strategy Figure gives the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use Figure , the power rule from calculus, to find the solution. We use Figure to calculate the average velocity of the particle. To determine the average velocity of the particle between 1.
The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use Figure and Figure to solve for instantaneous velocity. The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually Figure.
The reversal of direction can also be seen in b at 0. But in c , however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of x t is decreasing toward zero, becoming zero at 0. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations.
The graphs must be consistent with each other and help interpret the calculations. Speed is always a positive number. Check Your Understanding The position of an object as a function of time is. Related questions How do I determine the molecular shape of a molecule? What is the lewis structure for co2? What is the lewis structure for hcn? How is vsepr used to classify molecules? What are the units used for the ideal gas law?
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