The SI unit of time is the second. The SI unit of speed and velocity is the ratio of two — the meter per second. For a circle, the problem is simple: cos t , sin t will trace out a circle covering a constant amount of arc length per unit time. The analogous parameterization for an ellipse, a cos t , b sin t will move faster near the longer semi-axis and slower near the shorter one.
A differentiable curve is said to be regular if its derivative never vanishes. In words, a regular curve never slows to a stop or backtracks on itself.
Two differentiable curves and. Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The signed curvature of a curve parametrized by its arc length is the rate of change of direction of the tangent vector. The absolute value of the curvature is a measure of how sharply the curve bends. A curve traced out by a vector-valued function is parameterized by arc length if.
Such a parameterization is called an arc length parameterization. It is nice to work with functions parameterized by arc length, because computing the arc length is easy. The curvature of a straight line is zero. The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number. In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector.
It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. A unit normal vector of a curve, by its definition, is perpendicular to the curve at given point.
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 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 slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. 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. 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.
Calculate the instantaneous velocity given the mathematical equation for the velocity. Calculate the speed given the instantaneous velocity. 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. Using Equation 3. Calculate the average velocity between 1. Strategy Equation 3. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use Equation 3. We use Equation 3. To determine the average velocity of the particle between 1. What is the speed of the particle at these times? Strategy The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the 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 3. 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. Long answer: Any real measurement takes a finite amount of time and consists of discrete data points.
As such it is technically impossible to measure the instantaneous velocity. However, we can do well enough. A realistic measurement of velocity may be done as follows.
Now I need to justify to you why this can be "good enough" for a given experiment or calculation. Imagine you are running up and down a hill. The hill is a m long.
Then it will take you 20 s to run up the hill and 10 s to run down, 30 s total. First imagine they ONLY measure how long it takes you run the whole race. This measurement is clearly missing some details because you were never actually running 6. The average is just an estimate. Now imagine instead they measure your time at the top of the hill and at the bottom. Let's now imagine the person measures your velocity every second. However, they haven't gained any additional information over the previous case where they measured at the top and the bottom.
That is, they are making measurements faster than your velocity is changing!! This is the crux of my answer. Well that depends on whatever thing you are measuring.
For example, a runner probably doesn't change their pace very much over the course of 5 s so you could probably measure every 5 s and get a pretty good estimate of their instantaneous velocity.
But now consider you were trying to measure the velocity of a pinball in a pinball machine. The pinball's velocity can undergo many major changes in the course of 5 s. This means you should measure its velocity much more often!
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