Using the velocity-time graph, the acceleration of the object is determined to be 2 m/s 2 during the last five seconds of the object's motion. Thus, the slope (rise/run ratio) is (10 m/s)/(5 s) = 2 m/s 2. This is a total rise of +10 m/s and a total run of 5 s. Between 5 and 10 seconds, the line rises from 5 m/s to 15 m/s and runs from 5 s to 10 s. The slope of the line can be computed using the rise over run ratio. The positively sloped (i.e., upward sloped) section of the graph depicts a positive acceleration, consistent with the verbal description of an object moving in the positive direction and speeding up from 5 m/s to 15 m/s. The horizontal section of the graph depicts a constant velocity motion, consistent with the verbal description. Such a verbal description of motion can be represented by a velocity-time graph. In this part of Lesson 6, we will investigate the relationships between these two methods.Ĭonsider an object that moves with a constant velocity of +5 m/s for a time period of 5 seconds and then accelerates to a final velocity of +15 m/s over the next 5 seconds. Thus, there are now two methods available to solve problems involving the numerical relationships between displacement, velocity, acceleration and time. In Lesson 6, the focus has been upon the use of four kinematic equations to describe the motion of objects and to predict the numerical values of one of the four motion parameters - displacement (d), velocity (v), acceleration (a) and time (t). Thus, velocity-time graphs can be used to determine numerical values and relationships between the quantities displacement (d), velocity (v), acceleration (a) and time (t). In that Lesson, it was emphasized that the slope of the line on a velocity-time graph is equal to the acceleration of the object and the area between the line and the time axis is equal to the displacement of the object. The slope of this line will give us the acceleration.Lesson 4 of this unit at The Physics Classroom focused on the use of velocity-time graphs to describe the motion of objects. If the acceleration is constant the v-t graph will be a straight line. The slope of the tangent to a v-t graph at one point in time will give instantaneous acceleration in the case where there is non uniform (changing) acceleration. Using a method similar to deriving velocity from distance-time graphs, we can obtain values of acceleration from velocity-time graphs. 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

Graphical analysis of motion physics