Physics 121 for Majors

Transcription

1 Physics 121 for Majors Scheule Do Post-Class Quiz #3 Do Pre-Class Quiz #4 HW #2 is ue Wenesay Quiz #1 is ue Saturay, Sept. 16 Lab #1 is set up now, ue Monay Some Department Resources Computers N-212 ESC Computer room, available when class is not in session. N-361 Majors stuy room Tutorial Lab N-304 ESC Ethan Fletcher: M 1pm 3pm, W 7-9pm F 3pm 6-10 pm Spencer Vogel: M 1-4pm, W 1-5pm, F1-3 pm Last Class How physics works Observation Moeling Quantifying Getting meaningful answers Units Significant figures Estimation Derivatives Position Velocity Acceleration Doing Derivatives in Mathematica Class 3 Position, Velocity, an Acceleration Derivatives an Integrals Differential Equations Toay s Class Graphing position, velocity, acceleration Blinking light plots - relating the real worl to graphs Derivative rules Integrals Differential Equations in Mathematica 1

2 Section 1 Review Problems Estimation Problem In the real worl, it s useful to make eucate guesses. We on t always have exact numbers to put into equations an exact answers come out. Estimation Problem What is the mass of the Eyring Science Center? The ensity of water is 1000 kg/m 3. Estimation Problem What is the mass of the Eyring Science Center? Volume: (12 x 4) ~ 80 m 60 m 16 m ~ m 3 Mass if a soli block: m kg/m3 ~ kg But the builing must be about 5% soli (smaller upper floors, etc.) kg 0.05 ~ kg Section 2 Describing Motion Graphs an Motion It is very important to make connections between graphs an the real worl. Below is a graph of x vs t. Move your han so it matches the motion inicate in the graph. Now think of this as v (t ). Move your han so it matches the motion. 2

3 Blinking Light Plots Think of a spaceship with a light that blinks once each secon. The spaceship flies by a platform with a ruler so we can easily measure its position as the spaceship flies by. Blinking Light Plot 1 Blinking Light Plot 2 Vectors vs Scalars Scalars are just numbers, like temperature. Vectors are quantities that have size (magnitue) an irection, like velocity. In 3-D space, we nee three separate numbers to efine a vector In 1-D, we nee just one number Vectors in 1-D Vectors can be expresse in terms of a number like 15 m/s. Vectors can be expresse in terms of magnitue an irection: Magnitue is POSITIVE: 15 m/s Direction is just + or. It s -- in this example. Position Position is a vector but we ll stay in 1-D for now. We must choose an origin (the zero point) an ecie which irection is positive We must choose units -- This is easy! 3

4 Displacement Displacement is the change in position. We write it as x, y, etc. A note on notation: q = q f q i (final initial) or q = small bit of q Displacement can be positive, zero, or negative. -- This is a little harer... Graphing Graphs were (essentially) invente by René Descartes in the 17 th century. Cartesian coorinates are name after him. Graphing allows us to visualize relationships between variables an to treat them analytically using geometry. Case 1 Case 2 Section 3 A Rate Problem Consier the following ata: What is the overall value of T/ t? 15.5 o F/12 hr = 1.29 o F/hr Is T/ t the same at all times? What is T/ t at 6:00 am? 1.1 o F/1 hr = 1.1 o F/hr Time (hours) Temperature ( o F)

5 Consier the following ata: What is T/ t between 5:00 am an 7:00 am? o F/2 hr = o F/hr What is T/ t between 5:30 am an 6:30 am? o F/1 hr = o F/ hr What is T/ t between 5:54 am an 6:06 am? o F/.02 hr = 1.11 o F/ hr Time (hours) Temperature ( o F) A real experiment woul look more like this: What is T/ t between 5:00 am an 7:00 am? 2.3 o F/2 hr = 1.15 o F/hr What is T/ t between 5:30 am an 6:30 am? 1.2 o F/1 hr = 1.2 o F/hr What is T/ t between 5:54 am an 6:06 am? 0.1 o F/0.2 hr = 0.5 o F/hr Time (hours) Temperature ( o F) Calculating rates with experimental ata can be har! T(t) Plot Where is T/ t on the graph? Draw a sketch of T/ t. T/ Plot Draw a sketch of 2 T/ 2. 2 T/ 2 Plot Section 4 Graphing Velocity an Acceleration 5

6 Spee vs Velocity Spee is the magnitue of the velocity (+) Spee is a scalar Velocity is a vector, so 1-D, it can be positive, zero, or negative Velocity can be average ( x/ t) or instantaneous (x/) Spee is usually instantaneous, as average spee may be ambiguous. Case 1 Case 2 Section 5 Derivative Rules How Derivatives Work The important thing to remember is that erivatives can be thought of as either Rates (time) Slopes (on a graph) Recipes for how quantity A changes when quantity B changes How Derivatives Work f = lim f t + t f(t) t 6

7 How Derivatives Work Fin f where f t = t f t = t + t t = t + 2t t + t t t t 2t t + t = 2t + t = t f = lim f t = 2t Derivative Rules 1 A is a constant. A = 0 At = A At 2 At n = 2At = nat Derivative Rules 2 A an k are constants. A sin kt = Ak cos kt A cos kt = Ak sin kt Derivative Rules 3 Af(t) = A f(t) f t + g t = f t + g(t) f t g t = f(t) g t + g(t) f(t) f g f[g t ] = g Exercises 3 t + 2t = t t sin(kt) = sin(kt) + kt cos(kt) Section 6 Integrals sin(kt ) = cos kt 2kt 7

8 Unoing a erivative? Derivatives x(t) v(t) a(t) How o we go the other way x(t) v(t) a(t) We nee the inverse of a erivative! Integrals Are Anti-Derivatives If f an g are two functions of t such that f t = g(t) Then we can solve for g by: g t = f t Integrals to Memorize Af t + Bg t = A f t + B g t t = t n + 1, n 1 t = ln t sin(kt) = cos(kt) k cos(kt) = sin(kt) k Integrals Are Anti-Derivatives Let s try an example: f t = 4t g t = f t = 4 3 t So as avertise f t = g = 4 3 t = 4t Constant of Integration But note that if c is any constant an g t = 4 3 t + c Constant of Integration If we know the value of g at any specific time, such as g (0), we can etermine a unique value for the constant of integration. f t = g 4 3 t + c = 4t The same! So to uno a erivative with an integral, the general solution, g, must have a constant ae to the values of the integral of f. Let s look at a specific example to see how this works. 8

9 Constant of Integration The change in temperature between 7 am an 9 am is measure to be T = 4 /hr (t 7 hr) where t is the time in hours. The temperature at 7 am is 57 o F. Fin T (t ). T(t) = 2 /hr t 28 /hr t + C T 7 hr = 57 = C C = 155 T(t) = 2 /hr t 28 /hr t An Easier Way The change in temperature between 7 am an 9 am is measure to be T = 4 /hr t where t is the time in hours after 7am. The temperature at 7 am is 57 o F. Fin T (t ). T = 2 /hr t + C T 0 = 57 = C T = 2 /hr t + 57 Section 7 Velocity an Position from Acceleration Rates Derivatives with respect to time are calle rates. Velocity is rate of change of position. Acceleration is the rate of change of velocity. Jerk is the rate of change of acceleration (but it s almost never use). A Sample Acceleration As you press on the accelerator of your car, your car accelerates accoring to the function a t = At Where A = 3.25 m/s Convince yourself that this acceleration has the correct units. A Sample Acceleration Your car accelerates accoring to the function a t = At You re going own the highway at 25.0 m/s when you hit the accelerator. What is your velocity after 12.0 s of acceleration? How far have you travele in that time? 9

10 A Sample Acceleration a t = At A = m/s v 0 = 25.0 m/s x 0 = 0 m v = At = 1 3 At + c v 0 = c = 25.0 m/s x = 1 3 At + c = 1 12 At + ct + x 0 = = 0 m v 1.20 = 43.7 m/s x 1.20 = 356 m Differential Equation 1 a t = At Fin x(t) an v(t) if x(0)=0 an v(0)=25.0m/s Equations of motion: v t = x At = v Initial Conitions x 0 = 0 v 0 = v = 25.0 m/s Differential Equation 2 a t = At Fin x(t) an x (t) if x(0)=0 an x (0)=25.0m/s Equation of motion: At = x Initial Conitions x 0 = 0 x 0 = 25.0 m/s Section 8 Using Mathematica Derivatives in Mathematica y = a x^2 +b D[y,x] 2 a x x[t_]=4*t^2+7 D[x,t] 0 D[x[t],t] 8 t v=d[x[t],t] 8 t v[3] 8 t [3] v[t_]=d[x[t],t] 8 t v[3] 24 One Other Form y = a t^2 +b D[y,t] 2 a t y[t_]=a t^2 +b; y [t]=2 a t 10

11 Differential Equation 1 Differential Equation 2 Section 9 Spreasheet Solutions Spreasheets Useful for hanling numerical ata We ll use spreasheets for a few problems that can t be one easily with equations Basic manipulations are fairly easy Useful for graphing ata sets Excel (Microsoft Office) is available in computer labs throughout campus. Spreasheet Equations v = x a = v becomes becomes x = v t v = a t Start with initial values of x an v an calculate how they change after a tiny time Δt. Use the latest values of v an a to o that. Spreasheet Equations x = v t v = a t Start with initial values of x an v an calculate how they change after a tiny time Δt. Use the latest values of v an a to o that. Important! This is not very accurate unless Δt is very small! a Δt t x v Check a

12 Two Excel Tips I ll give you a template you can use in Excel. Excel is available on computers aroun campus in case you on t have it on your computer. In a new cell, type =, then click on the cells (from the previous row) you wish to use in your calculation along with the appropriate operators. Section 10 Recap Highlight cells, click on the lower right corner an rag own to repeat the cells. They ll automatically upate to use the cells from the correct lines. Big Ieas Using erivatives an integrals, we can obtain position, velocity, an acceleration if we know any one of them (plus initial conitions) Graphs an blinking light plots can help us unerstan motion Motion equations are typically ifferential equations 12