2.2 Average vs. Instantaneous Description

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1 2 KINEMATICS 2.2 Average vs. Instantaneous Description Name: 2.2 Average vs. Instantaneous Description Average vs. Instantaneous Velocity In the previous activity, you figured out that you can calculate average velocity as v avg = x f x i t f t i = x t Height of falling ball The x/ t way of writing this is using a short-hand notation using the capital Greek letter Delta ( ). It is used to indicate that you are talking about the difference in a quantity, for example T could refer to the temperature difference between day and night, ie., T = T day T night. In kinematics it may, e.g., refer to the difference in position ( x) or the elapsed time ( t). h [m] You also saw that the instantaneous velocity of an object equals the slope of the position vs. time (x vs. t) graph at that particular time. Consider the plot to the right of the height of a falling ball as a function of time (Note: vertical axis is height, horizontal axis is time.) 1. Is the slope of this graph constant? 2. Is the velocity of the object constant? 11

2 2.2 Average vs. Instantaneous Description 2 KINEMATICS 3. What is the average velocity from the time the ball is released (t = ) until it hits the ground? 4. Would the average velocity be useful for predicting where the ball is at different times? In other words is the average velocity all that we would need to predict the location at any point in time? Explain. If we zoom in on a portion of this graph, what we find is that on a small scale the curved shape of the graph gives way to a straight line: height straight line Height of falling ball h [m] This tells us something very important if we want to know the instantaneous velocity for an object whose velocity is not constant, we can just find the average velocity over a very small time interval. This revelation helped spur the development of calculus. What the above sentence is 12

3 2 KINEMATICS 2.2 Average vs. Instantaneous Description saying, is that in the limit of the time interval t going to, the average velocity v avg over that tiny time interval equals the instantaneous velocity v: x v = lim t t dx dt ẋ This is how a derivative is defined. In our case of velocity, it is the change in position over the change in time measured over an extremely small time interval. This makes velocity the derivative of position with respect to time. A derivative tells us what the slope of a function is at any point on the graph. The equation above introduced some more notation: You will learn the exact definition of the limit ( lim ) in the calculus part of the class, but for the most part it is what one intuitively thinks. The dx dt notation replaces the by d, making them so called differentials, that express that they are not just regular differences but very (infinitesimally) small differences now. Finally, ẋ is a physics-specific shorthand notation for a time derivative. In math, you may have seen d f dx f (x) before, where the ( prime ) indicates that you re now talking about the derivative of the function f, rather than f itself. The dot serves the same purpose, but it is used for derivatives with respect to time Average and Instantaneous Acceleration An object whose velocity is changing is said to be accelerating. Acceleration is defined as the rate of change in velocity with respect to time, and thus acceleration is related to velocity in the same way that velocity is related to position: a avg = v t v a = lim t x dv dt v 5. Your shiny new Porsche 911 Turbo S accelerates from to 1 km/h (that s about 6 mph) in 2.7 seconds. What is its average acceleration? How does it compare to the free-fall acceleration by gravity, g = 9.8 m/s 2 (more about free-fall later)? 13

4 2.2 Average vs. Instantaneous Description 2 KINEMATICS 6. If a function has a curved shape (that is, it looks like something other than a straight line), will its slope be the same everywhere (i.e., will x(t) have the same slope at every value of t)? Based on that, would you expect a function that has a curved shape to have a derivative that has the same value everywhere? Discuss and explain Derivatives of Polynomials Finding Instantaneous Velocity and Acceleration You know that the average velocity of an object is its net change in position over some time interval. We can this write out formally: v avg (t) = x t = x(t + t) x(t) (t + t) t = x(t + t) x(t) t From there, we get instantaneous velocity, which tells us exactly what the velocity is at any given moment in time by making t smaller and smaller (that s sometimes expressed as making t infinitesimally small. 7. Should the instantaneous and average velocities always be the same? Should they always be different? Discuss and explain. A derivative tells us what the slope of a function is at one specific location on its graph. If our graph is a function of time, that means the derivative would tell us precisely what the slope is at one instant in time. Therefore, the derivative of position with respect to time tells us the instantaneous velocity as as function of time. This means that we are looking at the change in position over the change in time for a very small time interval. So, we can find the instantaneous velocity by looking at 14

5 2 KINEMATICS 2.2 Average vs. Instantaneous Description how position changes over a time interval t, and make that time interval infinitesimally small. Effectively, the instantaneous velocity is the average velocity over an infinitesimally small time interval. v(t) = dx dt = lim x t t Consider the case of the falling ball, whose position is given by x(t) = 2m 4.9 m s 2 t2 8. What is the instantaneous velocity as a function of time? (use the definition using the limit above do not use the short-cut for taking a derivative. Hint: Wait until after dividing by t to take the limit of t.) 9. Let s consider a more general case though âăş a position function x(t) = A + Bt n, where A, B, and n are constants, and t is the variable that the position x(t) is a function of. Use the limit definition of a derivative to find the derivative of this general polynomial function. Keep in mind that for the derivative, we will want t to go to zero. Show your work clearly below. At this point, check with an instructor. 15

6 2.2 Average vs. Instantaneous Description 2 KINEMATICS 1. Through the previous problem, you should have found a general rule for taking the derivative of a polynomial. Use that general rule now to take the derivative of the height of the ball as a function of time, x(t) = 2m 4.9 m s 2 t 2. Do you get the same thing for the instantaneous velocity by using this rule for taking a derivative as when you used the limit definition? 11. What is the instantaneous acceleration of the falling ball? Explain your approach (i.e., what is acceleration qualitatively), and show your work. 12. Is the acceleration constant, or does it change in time? Note that this result is an important conclusion for objects in free fall. 16

7 2 KINEMATICS 2.2 Average vs. Instantaneous Description Graphical and Analytical Position, Velocity, Acceleration You measure the position of a car as it rolls up a slope and find it to be described by for the first 4 seconds of its motion. x(t) = 8 m s t 2m s 2 t2 13. Graph the position as a function of time. Then, describe the motion of the car in words. 1 Position vs time 8 x [m] By focusing on the slope of the position plot, try to plot the velocity as a function of time. 15. Do the same process to estimate what the acceleration will look like as a function of time. 1 Velocity vs time 4 Acceleration vs time 5 2 v [m/s] a [m/s 2 ]

8 2.2 Average vs. Instantaneous Description 2 KINEMATICS Instead of figuring out what the velocity and acceleration look like graphically, we can do it analytically by taking derivatives. 16. Using derivatives, what is the velocity of the car as a function of time? Does this match your graphical plot? 17. Using derivatives, what is the acceleration of the car as a function of time? Does this match your graphical plot? 18. Give an explanation below of the difference between the average and instantaneous velocity. 18

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