Figure 3. Given the position-versus-time graph of Figure , find the velocity-versus-time graph. Notice that the object comes to rest instantaneously, which would require an infinite force. Thus, the graph is an approximation of motion in the real world. 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.
Show Answer. Time interval 0. Time interval 1. 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.
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.
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 velocity of the particle gives us direction information, indicating the particle is moving to the left west or right east. If the horizontal direction of the ball is defined as the positive x direction, and vertically upward is defined as the positive y direction, then the magnitudes of the x and y components of the instantaneous velocity are:. Toggle navigation. Instantaneous Velocity Formula. Instantaneous Velocity Formula Velocity is a measure of how quickly an object moves from one position to another.
Instantaneous Velocity Formula Questions: 1 A cat that is walking toward a house along the top of a fence is moving at a varying velocity. Answer: The cat's velocity can be found using the formula: The cat's position has only one component, since it is moving in a straight line along the fence. Thus, the zeros of the velocity function give the minimum and maximum of the position function. Time interval 0. Time interval 1. The graph of these values of velocity versus time is shown in Figure 3.
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 x t. If each term in the x t x t equation has the form of A t n A t n where A A is a constant and n n is an integer, this can be differentiated using the power rule to be:. Note that if there are additional terms added together, this power rule of differentiation can be done multiple times and the solution is the sum of those terms.
The following example illustrates the use of Equation 3. As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4. Skip to Content Go to accessibility page.
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