1. AP Physics ST Equations of Motion Free Fall

2. Equations of Motion EVERYTHING MOVES… relatively speaking! The two most commonly asked questions regarding motion… 1.HOW FAST? 2.HOW FAR? maineoutdoorjournal.mainetoday.com

3. Equations of Motion These two questions help explain… – WHY 2 are in terms of velocity and 2 are in terms of displacement versus time. – WHY there are 4 total equations… each lacking 1 of 4 descriptors of motion. maineoutdoorjournal.mainetoday.com

4. Equations of Motion If the acceleration of an object varies with time, like the space shuttle launching, the motion can become quite complicated. Very common to simplify motion by considering the acceleration to be constant… we’ll stick with constant acceleration! archives.sensorsmag.com

5. Equations of Motion Equation #1 Recall: – The slope of a line on a v-t graph shows acceleration. Noting our tendency to ask “how fast” motivates us to solve for v f … vfvf vivi titi tftf

6. Equations of Motion Equation #2 Consider now the area under the curve… – Trapezoid – Triangle and rectangle

7. Equations of Motion Equation #2 Trapezoid first… – Area of a trapezoid: – Physical substitution…

8. Equations of Motion Equation #3 Triangle and Rectangle… – Area of a triangle plus area of a rectangle… – Physical substitution…

9. Equations of Motion Equation #3 For the purpose of “lacking 1 of 4 descriptors of motion” make the following substitution…

10. Equations of Motion Equation #3 Simplifying the substitution yields…

11. Equations of Motion Equation #4 Again for the purpose of “lacking 1 of 4 descriptors of motion” square the first equation and simplify…

12. Equations of Motion Equations of MotionLacking * * *most meaningful of the 4 equations of motion.

13. Equations of Motion Derived with Calculus Consider only motion along a straight line… – Constant acceleration. – Slope of a line on a v-t graph shows acceleration. The defining equation of acceleration is… v t

14. Equations of Motion Derived with Calculus Acceleration… can be rewritten as… or in its integral form…

15. Equations of Motion Derived with Calculus Noting constant acceleration, “a” can be removed from integral… C 1 is known as the initial condition which in this case is the initial velocity. v t

16. Equations of Motion Derived with Calculus C 1 represents the initial condition of motion. Let v = v i when t = 0… Representing v as v f and substituting C 1 yields …

17. Equations of Motion Derived with Calculus Now consider the defining equation for velocity is…

18. Equations of Motion Derived with Calculus Velocity… can be rewritten as… or in its integral form…

19. Equations of Motion Derived with Calculus Note that the time- varying velocity has already been defined for constant acceleration… Making this substitution…

20. Equations of Motion Derived with Calculus Simplifying integral…

21. Equations of Motion Derived with Calculus To determine C 2, let x = x i at t = 0 and solve… Representing x as x f and substituting C 2 yields …

22. Free Fall Free fall – An object moving freely under the influence of gravity alone. Magnitude of gravity (g) at the surface of the earth is … patdollard.com

23. Free Fall Considering the magnitude of gravity to remain constant the equations of motion for objects in free fall become…

24. Lesson Summary

26. Example #1 Problem 2-30 A car is approaching a hill at 30.0 m/s when its engine suddenly fails, just at the bottom of the hill. The car moves with a constant acceleration of -2.00 m/s 2 while coasting up the hill. a.Write equations for the position along the slope and for the velocity as functions of time, taking x = 0 at the bottom of the hill, where vi = 30.0 m/s. b.Determine the maximum distance the car travels up the hill.

27. Example #2 Problem 2-60 A motorist drives along a straight road at a constant speed of 15.0 m/s. Just as she passes a parked motorcycle cop, the cop starts to accelerate at 2.00 m/s 2 to overtake her. Assuming the cop maintains this acceleration, a.Determine the time it takes the cop to reach the motorist. b.Find the speed of the cop as the cop overtakes the motorist. c.Find the total displacement of the cop as the cop overtakes the motorist.

28. Example #3 Problem 2-59 A test rocket is fired vertically upward from a well. A catapult give it an initial velocity of 80.0 m/s at ground level. Subsequently, its engines fire and it accelerates upward at 4.00 m/s 2 until it reaches an altitude of 1000 m. At that point its engines fail, and the rocket goes into free fall, with an acceleration of -9.8 m/s 2. a.How long is the rocket in motion above the ground? b.What is its maximum altitude? c.What is its velocity just before it collides with the Earth?