V inst = v or, using the slope equation we calulte the slope of the tangent (the red line): To obtain the instantaneous velocity, that is, the velocity at one instant (one point in time - say at exactly t 1), one must take the slope of the tangent line that just touches the curve at that point. This gives us the average velocity between the time interval from t 1 -to- t 2 Slope (of secant line) = Delta d / Delta t The slope of the secant line (the line that cuts the curve at the two intersecting points (d 1,t 1 and d 2, t 2) can be calculated by the usual slope method: To obtain the average velocity between two points in time (say t 1 and t 2) we can draw the secant line between these two points and calculate the slope of the secant line. It is sufficient to say, however that we can still obtain some useful information by relying on graphical analysis techniques. To analyze this function properly one would need to take the first derivative of the function using Calculus. We know that the velocity is changing as time goes on because the slope of this line is not constant and the function (of the form y = ax 2 + bx + c) is an increasing function. This graph illustrates the relationship between the position and the time for an object whose velocity is changing with time. Uniformly Accelerated Motion - Variable Speed We note that "m", the slope of the line (of the form, y = mx + b) is constant and can be calculated by several methods.ī. The slope m of the position-time graph gives the velocity of the object even when the relation between the position (d) and the time (t) is not a straight line. This is a typical graph of the relationship between position and time for an object moving at constant speed. Uniform Motion - Constant Speed - no acceleration (c) Acceleration gradually declines to zero when velocity becomes constant.A. The slope of this graph is acceleration it is plotted in the final graph. (b) The velocity gradually approaches its top value. (a) The slope of this graph is velocity it is plotted in the next graph. This motion begins where the motion in ends. Graphs of motion of a jet-powered car as it reaches its top velocity. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward. If we call the horizontal axis the x x size 12, after which time the slope is constant. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. Slopes and General Relationshipsįirst note that graphs in this text have perpendicular axes, one horizontal and the other vertical. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate one-dimensional kinematics. Graphs not only contain numerical information they also reveal relationships between physical quantities. time.Ī graph, like a picture, is worth a thousand words. Determine average or instantaneous acceleration from a graph of velocity vs.Determine average velocity or instantaneous velocity from a graph of position vs.Describe a straight-line graph in terms of its slope and y-intercept.
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