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Should be : Wednesday, Sept 8

Should be : Wednesday, Sept 8. Announcements. You may attend any of the instructors’ office hours. List maintained on Blackboard under “Staff / Office Hours”. HW #1 due tonight at 11:59 p.m. HW #2 now available, due Thurs Sept 9 at 11:59 p.m.

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Should be : Wednesday, Sept 8

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  1. Should be: Wednesday, Sept 8

  2. Announcements • You may attend any of the instructors’ office hours. List maintained on Blackboard under “Staff / Office Hours”. • HW #1 due tonight at 11:59 p.m. • HW #2 now available, due Thurs Sept 9 at 11:59 p.m. • Solutions to HW #1 will be available on Blackboard (under “Solutions”) after the due date/time. • Quiz grades will be posted to Blackboard after your recitation instructor has graded the quizzes (also returned next week).

  3. Last Time: Motion in One Dimension Displacement, Velocity, Acceleration • Today: One-Dimensional Motion with Constant Acceleration Freely Falling Objects (under gravity)

  4. Constant/Uniform Acceleration • Constant/Uniform Acceleration: magnitude and direction of the acceleration does not change. • Constant acceleration is important, because it applies to many natural phenomena. • One such example is objects in “free fall” near the surface of the Earth. Neglecting air resistance, all objects (independent of mass) “fall” with the same downward acceleration !

  5. Constant Acceleration • Key Point: • Under constant acceleration, the instantaneous acceleration at any point in a time interval is equal to the average acceleration over the entire time interval. This Means: velocity slope = a v at Let: ti = 0, vi = v0, tf = t, and vf = v v0 v0 time t (for constant a only)

  6. Constant Acceleration velocity velocity a < 0 a > 0 v v0 time time Δt Average velocity in any time interval Δt is : Just the average of the velocities at the start/finish of the time interval ! (for constant a only)

  7. Constant Acceleration Recall: The displacement Δx and average velocity v Set: If ti = 0 : (for constant a only) Using v = v0 + at : (for constant a only)

  8. Constant Acceleration velocity slope = a v For constant acceleration, the displacement Δx is just equal to the area under the velocity-vs-time graph ! at v0 v0 time t Recall: Area of Triangle = ½ (base) (height)

  9. Constant Acceleration Finally: Recall and Plugging in for t gives us: (for constant a only)

  10. “Equations of Motion”for Constant Acceleration Information Equation Velocity as a function of time Displacement as a function of time Velocity as a function of displacement • These are for motion along a single axis (x-axis) with acceleration alongthat direction. • If acceleration and motion along y-axis, just change x  y. • Assumes that at t = 0, v = v0. • Recall: Δx = x – x0

  11. Example A car starting from rest undergoes a constant acceleration of a = 2.0 m/s2. What is the velocity of the car after it has traveled 100 m? How much time does it take to travel the 100 m?

  12. Example: Problem 2.27 A car traveling east at 40.0 m/s (= 89.48 mph !) passes a trooper hiding at roadside. The driver uniformly reduces his speed to 25.0 m/s in 3.50 s. What is the magnitude and direction of the car’s acceleration as it slows down? How far does the car travel during the 3.50-s time period? v0 = 40 m/s West East -x +x

  13. Example: Problem 2.29 A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final velocity of 2.80 m/s. Find the truck’s original speed. Find its acceleration.

  14. Freely Falling Objects • Key Point: • When air resistance is negligible, all objects falling under the influence of gravity near the Earth’s surface fall at the same constant acceleration. Aristotle (384 – 322 B.C.) Plato (428 – 348 B.C.) Heavier objects fall faster than lighter objects.

  15. Galileo Galilei (1564-1642) • Italian physicist and philosopher • Professor at the University of Padua • Invented the thermometer and the pendulum clock “Galileo, perhaps more than any other single person, was responsible for the birth of modern science.” – Stephen Hawking

  16. Galileo Galilei (1564-1642) • First to observe the heavens with a telescope • Defended the (disturbing) idea of Copernicus (1473-1543) that the Earth was not the center of the universe • Tried for heresy by Catholic Church (1992: Vatican found him not guilty) • Undertook series of experimental studies to describe the motion of bodies: mechanics

  17. Freely Falling Objects • What is a freely falling object ? • Any object moving freely under the influence of gravity alone, regardless of its initial motion. • Denote magnitude of free-fall acceleration as: • g = 9.80 m/s2(magnitude on the surface of the Earth) • If we: • Neglect air resistance • Assume acceleration doesn’t vary with altitude (over short vertical distances) “up” = +y Motion of freely falling object is the same as one-dimensional motion with constant acceleration !

  18. +y = “up” “Equations of Motion”for Freely Falling Objects Information Equation Velocity as a function of time Displacement as a function of time Velocity as a function of displacement Recall: Δy = y – y0

  19. Objects Rising Against Gravity If object’s initial (upward) velocity is v0 , how high will it rise? y y = 0

  20. Objects Rising Against Gravity If object’s initial (upward) velocity is v0 , how high will it rise? First, solve for the time to reach the highest point.  At highest point, v = 0 ! y Second, calculate the maximum height after solving for the time to reach the highest point. Alternative method … y = 0

  21. Example: Multiple Choice #10 (p. 48) A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground … True or False: The velocities of the two balls are equal. y v0 y = h v0 h y = 0

  22. Example: Multiple Choice #10 (p. 48) A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground … True or False: The speed of each ball is greater than v0. y v0 y = h v0 h y = 0

  23. Example: Multiple Choice #10 (p. 48) A student at the top of a building throws a red ball upward with speed v0, and then throws a green ball downwards with the same initial speed v0. Immediately before the balls hit the ground … True or False: The acceleration of the green ball is greater than that of the red ball. y v0 y = h v0 h y = 0

  24. Example: Problem 2.50 A small mailbag is released from a helicopter that is descending steadily at 1.50 m/s. After 2.00 s: What is the speed of the mailbag? ; and How far is it below the helicopter ? How do (a) and (b) change if the helicopter is rising steadily at 1.50 m/s?

  25. Reading Assignment • Next class: 3.1 – 3.3

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