Position-versus-Time Graphs Below is a motion diagram, made at 1 frame per minute, of a student walking to school. A motion diagram is one way to represent the student s motion. Another way is to make a graph of x versus t for the student: Slide 1-65
QuickCheck Which position-versus-time graph represents the motion shown in the motion diagram?
Example 1 - Bob leaves home at 9:05 and runs at a constant speed to the lamppost. He reaches the lamppost at 9:07, immediately turns and runs to the tree. Bob arrives at the tree at 9:10. Lamppost Home Tree 0 200 400 600 800 1000 1200 x (yards) Draw a position-time diagram. Find the average velocity for each part of the run and for the entire run.
Interpreting a Position Graph Slide 1-66
QuickCheck Here is a position graph of an object: At t = 1.5 s, the object s velocity is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. 10 m/s. Slide 2-48
QuickCheck Here is a position graph of an object: At t = 3.0 s, the object s velocity is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. 10 m/s. Slide 2-50
QuickCheck How would you interpret this position-time plot? A. An object heads in the positive x-direction with non-uniform speed for some time but then heads back in the negative x-direction with uniform speed. B. An object heads in the positive x-direction for the entire journey but with uniform speed that varies and stopping once for 4 seconds. C. This position-time is not physically possible for an object.
Uniform Motion If you drive your car at a perfectly steady 60 mph, this means you change your position by 60 miles for every time interval of 1 hour. Uniform motion is when equal displacements occur during any successive equal-time intervals. Uniform motion is always along a straight line. Riding steadily over level ground is a good example of uniform motion. Slide 2-20
Uniform Motion An object s motion is uniform if and only if its position-versus-time graph is a straight line. The average velocity is the slope of the positionversus-time graph. The SI units of velocity are m/s. Slide 2-21
The Mathematics of Uniform Motion Consider an object in uniform motion along the s-axis, as shown in the graph. The object s initial position is s i at time t i. At a later time t f the object s final position is s f. The change in time is t t f t i. The final position can be found as: Slide 2-27
Instantaneous Velocity An object that is speeding up or slowing down is not in uniform motion. (i.e. the position-time graph is not a straight line) We can determine the average speed v avg between any two times separated by time interval t by finding the slope of the straight-line connection between the two points. The instantaneous velocity is the object s velocity at a single instant of time t. The average velocity v avg s/ t becomes a better and better approximation to the instantaneous velocity as t gets smaller and smaller. Slide 2-31
Instantaneous Velocity Motion diagrams and position graphs of an accelerating rocket. Slide 2-32
Instantaneous Velocity As t continues to get smaller, the average velocity v avg s/ t reaches a constant or limiting value. The instantaneous velocity at time t is the average velocity during a time interval t centered on t, as t approaches zero. In calculus, this is called the derivative of s with respect to t. Graphically, s/ t is the slope of a straight line. In the limit t 0, the straight line is tangent to the curve. The instantaneous velocity at time t is the slope of the line that is tangent to the position-versus-time graph at time t. Slide 2-33
Finding Velocity from Position Graphically Slide 2-36
Finding Velocity from Position Graphically Slide 2-37
QuickCheck Here is a motion diagram of a car moving along a straight road: Which position-versus-time graph matches this motion diagram? Slide 2-40
QuickCheck 2.4 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? E. None of the above. Slide 2-42
QuickCheck Which velocity-versus-time graph goes with this position graph? Slide 2-61
QuickCheck When do objects 1 and 2 have the same velocity? A. At some instant before time t 0. B. At time t 0. C. At some instant after time t 0. D. Both A and B. Slide 2-52
Example 2
Finding Position from Velocity Suppose we know an object s position to be s i at an initial time t i. We also know the velocity as a function of time between t i and some later time t f. Even if the velocity is not constant, we can divide the motion into N steps in which it is approximately constant, and compute the final position as: The curlicue symbol is called an integral. The expression on the right is read, the integral of v s dt from t i to t f. Slide 2-54
Finding Position From Velocity The integral may be interpreted graphically as the total area enclosed between the t-axis and the velocity curve. The total displacement s is called the area under the curve. Slide 2-55
QuickCheck Here is the velocity graph of an object that is at the origin (x 0 m) at t 0 s. At t 4.0 s, the object s position is A. 20 m. B. 16 m. C. 12 m. D. 8 m. Slide 2-56
QuickCheck Here is the velocity graph of an object that is at the origin (x 4 m) at t 0 s. At t 4.0 s, the object s position is A. 20 m. B. 16 m. C. 12 m. D. 8 m. Slide 2-56
Example 3 - The figure below shows the velocity graph for a particle having initial position x o = 10 m at t 0 = 0 s. Draw a position-time graph for the particle. At what time or times is the particle found at x = 45 m? Does the graph have a turning point?