New! A new student section has been written called Calculus from 30,000 feet. Borrowed from Steven Strogatz's book Infinite Powers, this addendum is meant to be used after the AP exam and gives students a greater insight into how the Infinity Guideline breaks down comples probems into simper ones and then reassembles them. No homework or problems - just an attempy to see the Calculus forest from the trees.
Straight Line Motion Revisited Homework Answersl
To find the observable consequences of this curvature we need to take another step. When spacetime is flat freely moving objects obey Newton's first law i.e. they move in a straight line at constant speed. When spacetime is curved freely moving objects obey a different equation called the geodesic equation:
But all this is a bit abstract, and I suspect you're after a more intuitive feel for how does spacetime curvature cause straight lines not to be straight? Well the most obvious definition of a straight line is the path of a light ray, because we all learn in school that light travels in straight lines. And the obvious demonstration of this is the bending of light rays by the Sun.
If we shine a light ray so that it just grazes the Sun's surface then that light doesn't travel in a straight line. We can calculate the deflection using the geodesic equation and the values of the Cristoffel symbols, and we find the light ray is bent by about $1.75$ arcseconds. But this is a tiny, tiny amount. $1.75$ arcseconds is about the angle subtended by a baseball at a distance of $9$ kilometers - you couldn't even see a baseball $9$ km away!
11. Consider two cylinders that start down identical inclines from rest except that one is frictionless. Thus one cylinder rolls without slipping, while the other slides frictionlessly without rolling. They both travel a short distance at the bottom and then start up another incline. (a) Show that they both reach the same height on the other incline, and that this height is equal to their original height. (b) Find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes to reach the height on the second incline. (c) Explain why the time for the rolling motion is greater than that for the sliding motion.
An object at rest (not moving) will remain at rest unless acted on by an unbalanced force. Also, an object in motion will continue to move at a constant speed in a straight line unless acted on by an unbalanced force.
In 2-d projectile motion without air resistance the equation relating the velocity ##\vec v_\!f## at time ##t_\!f## and the initial velocity ##\vec v_0## is $$\vec v_\!f=\vec v_0+\vec gt_\!f.$$The three vectors participating in the equation form a closed triangle the sides of which are ##OA = v_0##, ##OF = v_\!f## and a third vertical side ##AF = g t_\!f## (see figure 1). Triangle OAF in the figure corresponds to a projectile that is at positive vertical displacement ##\Delta y## at time ##t_\!f## after reaching maximum height. I call this the velocity triangle; its geometric construction from three given lengths is outlined in Appendix I. Triangle OAB is another velocity triangle corresponding to the projectile returning to launch level and is drawn for later reference.
Beyond being a curiosity, the position triangle can serve as a useful tool for solving problems that cannot be addressed by a velocity triangle construction. Consider the problem in which a projectile is fired up a plane inclined at angle ##\alpha## relative to the horizontal. The projection angle is ##\beta## relative to the incline and we are seeking the landing distance ##d## up the incline. The traditional strategy is to find the trajectory parabola by eliminating ##t_\!f## from the horizontal and vertical position equations, find the horizontal distance ##x_\!f## at which the parabola intersects the straight line ##y = x \tan\alpha##, find the final height ##y_\!f## and finally use the Pythagorean theorem to find the required distance ##d##. This two-page solution can be reduced to two lines using the position triangle.
3.4 Velocity & Other Rates of Change\n \n \n \n \n "," \n \n \n \n \n \n AP Physics Monday Standards: 1)a. Students should understand the general relationships between position velocity & acceleration for a particle.\n \n \n \n \n "," \n \n \n \n \n \n Projectiles Horizontal Projection Horizontally: Vertically: Vertical acceleration g \uf0af 9.8 To investigate the motion of a projectile, its horizontal and.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/24 Warm up 1. Do in notebook Estimate the instantaneous rate of change at x = 3 for the function by picking values close to 3.\n \n \n \n \n "," \n \n \n \n \n \n SECTION 4-4 A Second Fundamental Theorem of Calculus.\n \n \n \n \n "," \n \n \n \n \n \n 4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA \u2013 AP CALCULUS.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 1. Do in notebook. Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : warm-up Go over homework homework quiz Notes.\n \n \n \n \n "," \n \n \n \n \n \n Agenda 9\/23\/13 Hand in Great Graphing homework Quiz: Graphing Motion Rearranging equations practice Discuss homework p. 44, p. 49 Notes\/ Discussion: Kinematic.\n \n \n \n \n "," \n \n \n \n \n \n Velocity - time graph 1. The velocity \u2013 time graph shows the motion of a particle for one minute. Calculate each of the following. (a) The acceleration.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/26 Warm up 1. Do in notebook True or False, if false explain why or give example 1. If, then 2. If, then 3. If, then 4. If, then.\n \n \n \n \n "," \n \n \n \n \n \n Warm-up 8\/31. Finding the derivative and calculating the derivative at a value.\n \n \n \n \n "," \n \n \n \n \n \n \uf09e The derivative of a function f(x), denoted f\u2019(x) is the slope of a tangent line to a curve at any given point. \uf0a1 Or the slope of a curve at any given.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/25 Warm up 1. Do in notebook Expand the binomials.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 9\/25 Finalize your letter to the MANufacture. Be ready to shared.\n \n \n \n \n "," \n \n \n \n \n \n 2.1 Position, Velocity, and Speed 2.1 Displacement \uf044 x \uf0ba x f - x i 2.2 Average velocity 2.3 Average speed \uf0ba \uf0ba\n \n \n \n \n "," \n \n \n \n \n \n 2.2 Differentiation Techniques: The Power and Sum-Difference Rules 1.5.\n \n \n \n \n "," \n \n \n \n \n \n Warm Up 10\/13 Simplify each expression. 16, (3 2 )\n \n \n \n \n "," \n \n \n \n \n \n Warm up 9\/08 1. Factor 2. Solve by Factor. Be seated before the bell rings DESK homework Warm-up (in your notes) Ch 5 test tues 9\/15 Agenda: Warmup Go.\n \n \n \n \n "," \n \n \n \n \n \n 3.2 Notes - Acceleration Part A. Objectives \uf070 Describe how acceleration, time and velocity are related. \uf070 Explain how positive and negative acceleration.\n \n \n \n \n "," \n \n \n \n \n \n Warmup. Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : go over hw Hw quiz Notes lesson 3.2.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 10\/16 (glue in). Be seated before the bell rings DESK homework Warm-up (in your notes) Agenda : go over hw Finish Notes lesson 4.5 Start 4.6.\n \n \n \n \n "," \n \n \n \n \n \n Warmup 9\/18 Use 1 st derivative test to determine maximum and minimums.\n \n \n \n \n "," \n \n \n \n \n \n 5.3: Position, Velocity and Acceleration. Warm-up (Remember Physics) m sec Find the velocity at t=2.\n \n \n \n \n "," \n \n \n \n \n \n Instantaneous Rate of Change The instantaneous rate of change of f with respect to x is.\n \n \n \n \n "," \n \n \n \n \n \n A car is moving along Highway 50 according to the given equation, where x meters is the directed distance of the car from a given point P at t hours. Find.\n \n \n \n \n "," \n \n \n \n \n \n 3023 Rectilinear Motion AP Calculus. Position Defn: Rectilinear Motion: Movement of object in either direction along a coordinate line (x-axis, or y-axis)\n \n \n \n \n "," \n \n \n \n \n \n Warm up 9\/23 Solve the systems of equations by elimination.\n \n \n \n \n "," \n \n \n \n \n \n February 6, 2014 Day 1 Science Starters Sheet 1. Please have these Items on your desk. I.A Notebook 2- Science Starter: Two Vocabulary Words: Motion Speed.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 8\/19 Warm up 1. Do in notebook Explain why these are incorrect :\n \n \n \n \n "," \n \n \n \n \n \n Ch. 8 \u2013 Applications of Definite Integrals 8.1 \u2013 Integral as Net Change.\n \n \n \n \n "," \n \n \n \n \n \n Warmup 9\/17 Mr. Xiong\u2019s car passes a stationary patrol cars at 65 mph. He passes another patrol car 8 min later going 55 mph. He was immediately stop and.\n \n \n \n \n "," \n \n \n \n \n \n 1.A Function and a Point 2.Equation of a line between 2 points 3.A point on the graph of a function 4.Information from a function\u2019s graph 5.Symbols on.\n \n \n \n \n "," \n \n \n \n \n \n Warm up 10\/15. Review: 1)Explain what is the FTC # 2. 2)Explain how to do each of these three problems a) b)C)\n \n \n \n \n "," \n \n \n \n \n \n Warm-upWarm-up 1.Find all values of c on the interval that satisfy the mean value theorem. 2. Find where increasing and decreasing.\n \n \n \n \n "," \n \n \n \n \n \n 3-4 VELOCITY & OTHER RATES OF CHANGE. General Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative! Ex 1a)\n \n \n \n \n "," \n \n \n \n \n \n Motion Graphs Learning Target: Be able to relate position, velocity, and acceleration quantitatively and qualitatively.\n \n \n \n \n "," \n \n \n \n \n \n Speed vs. Velocity.\n \n \n \n \n "," \n \n \n \n \n \n Bell Work 12\/1\/14 Pick up a Topic Reinforcement page from the front table. Take a seat and this week into your agenda. Begin working quietly on the Topic.\n \n \n \n \n "," \n \n \n \n \n \n Warm up Warm up 1. Do in notebook\n \n \n \n \n "," \n \n \n \n \n \n Ch.5, Sec.1 \u2013 Measuring Motion\n \n \n \n \n "," \n \n \n \n \n \n What is Motion?.\n \n \n \n \n "," \n \n \n \n \n \n Motion Review Challenge\n \n \n \n \n "," \n \n \n \n \n \n Lecture 2 Chapter ( 2 ).\n \n \n \n \n "," \n \n \n \n \n \n Motion and Force A. Motion 1. Motion is a change in position\n \n \n \n \n "," \n \n \n \n \n \n Graphing Motion Walk Around\n \n \n \n \n "," \n \n \n \n \n \n Take out your homework from Friday (only front should be completed)\n \n \n \n \n "," \n \n \n \n \n \n A function F is the Antiderivative of f if\n \n \n \n \n "," \n \n \n \n \n \n 2.3B Higher Derivatives.\n \n \n \n \n "," \n \n \n \n \n \n The Kinematics Equations\n \n \n \n \n "," \n \n \n \n \n \n Warm-up Enter the two functions into the y = in your\n \n \n \n \n "]; Similar presentations 2ff7e9595c