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Understanding Free Fall and Two-Dimensional Motion in Physics

Explore acceleration due to gravity, free-falling bodies, and 2-D motion with examples and concepts from physics lectures. Learn about Galileo's experiments and the kinematic quantities describing motion.

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Understanding Free Fall and Two-Dimensional Motion in Physics

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  1. Lecture 7 > Free Fall > 2-D Motion Giancoli, Physics 6/e Serway & Vuille, College Physics 8/e Tippens, Physics 7/e Young, Freedman & Ford, University Physics 14/e 1 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  2. Acceleration due to Gravity Objects reach the ground at the same time when dropped from the same height > An approximate example of constant acceleration is when a body falls to the ground from a height above the Earth. > Galileo Galilei, according to the myth, went to the top of the leaning tower of Pisa, dropped spheres made of different stuff, and noticed that these reached the ground at the same time > For heights close to the surface of the Earth, a falling body's acceleration is approximately 9.80 m/s2, or 32 ft/s2. + This is the case regardless of how high the body was located from the ground + or of what the initial velocity of the body was. 2 Villacorta--FEU-Phys1BLec-L07-1819Sem02 http://www.scientus.org/Galileo-Myths.html

  3. Freely Falling Bodies > When a body is in free fall, only the acceleration due to gravity is acting on it. + No other acceleration is present. + The acceleration due to gravity is always directed toward the ground. + Under free fall, bodies slow down as they go up & speed up as they fall down. Apollo 15 Free Fall Expt w David Scott Hammer (l) & Feather (r) dropped from the same height, fall at the same rate, reach the ground at the same time No air resistance since there's no air on the moon. 3 Villacorta--FEU-Phys1BLec-L07-1819Sem02 https://en.wikipedia.org/wiki/Free_fall

  4. Freely Falling Bodies contd > When a body is given an initial upward velocity, it will reach a maximum height before falling down. + At the highest point, the body is momentarily at rest: v = 0. + A body's speed before reaching the maximum height is the same as its speed after passing the maximum height for the same increment of time. (The velocities at these times are oppositely directed.) + If it takes some time to go up from a point to maximum height, then it will also take the same amount of time to go down from maximum height to the same point. Ball thrown straight up Highest pt: zero velocity 1s before & after highest pt: same height/y-position same velocity magnitude opposite velocity direction 2s before & after highest pt & 3s before & after higest pt: same description as above y-position v = 0 time 6s 0s 1s 2s 3s 4s 5s 4 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  5. Bodies in Motion > A body moving along a straight line can be described through + its displacement, + its velocity, and + its acceleration. > These kinematical quantities still describe the body's motion whether in 1-D motion (along a straight line) or in 2-D motion (along a flat surface). > Displacement, velocity, and acceleration can be generalized to describe a body's 2-D motion by breaking it down to 1-D. + Each dimension will have its own kinematical quantities. + For each dimension, there will be three quantities (x, v, a). + Thus, there will be six kinematical quantities that describe the object. 5 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  6. Bodies in Motion contd > Two-dimensional motion can be easily analyzed by considering the 1-D motion of the body twice. + The horizontal & vertical motions will be considered instead of the whole motion of the body. + On the x-y plane, the motion is broken into motions along the x-axis & y-axis. Getting the x- and y- components 6 https://www.physicsclassroom.com/class/vectors/Lesson-1/Independence-of-Perpendicular-Components-of-Motion Villacorta--FEU-Phys1BLec-L07-1819Sem02

  7. Bodies in Motion contd > The displacement, velocity, & acceleration of a body can each be broken into two components: + horizontal (x) components: x, vx, ax. + vertical (y) components: y, vy, ay. > These components can be obtained by the common decomposition of vectors. + Draw the vector quantity on a Cartesian plane. (Do NOT draw displacements, vectors, & accelerations on the same plane.) + Determine the sign of the components from their position on the plane. + Compute for the components using trigonometry. 7 https://www.physicsclassroom.com/class/vectors/Lesson-1/Independence-of-Perpendicular-Components-of-Motion Villacorta--FEU-Phys1BLec-L07-1819Sem02

  8. Bodies in Motion contd > Ex. A body at rest begins to move with a constant velocity of 4.00 m/s, 30º above the +x-axis. (a) What are the velocity components along the x- & y-axes? (b) What will be the x-displacement of the body after 2.00 s? (c) What will be the y-displacement of the body after the same interval of time? (d) What will be the displacement of the body after 2.00 s? 8 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  9. Free Fall in 2-D > Bodies in free fall can also undergo 2-D motion. + When the body is given a horizontal velocity before falling, the body will NOT fall in a straight line. + The horizontal velocity remains as the body falls since it is NOT affected by the downward pull of gravity. + The magnitude & direction of this horizontal velocity also remains the same since it is NOT affected by the vertical acceleration acting on the falling body. > The body will trace a parabolic path as it falls to the ground. 9 https://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontally-Launched-Projectiles-Problem-Solving Villacorta--FEU-Phys1BLec-L07-1819Sem02

  10. Free Fall in 2-D contd > Consider an object at a height h above the ground. It starts to fall after being given an initial horizontal velocity of v0. 10 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  11. 2-D Motion and Free Fall > Bodies moving in two dimensions can be described by breaking down the motion along the x- and y-axes. + The displacement, velocity & acceleration can be decomposed into components along the x- and y-axes. + Like in the 1-D case with constant acceleration, a body's motion through time can be determined by specifying the initial conditions. > Freely falling objects with a initial non-zero horizontal velocity show a motion that is similar to the 1-D free fall case. + The acceleration due to gravity only affects the horizontal motion, making the object fall to the ground faster and faster. + The horizontal motion remains the same due to the constant horizontal velocity that was initially applied. 11 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  12. Summary > Acceleration due to gravity, 9.80 m/s2. > A body's motion can be broken into two perpendicular motions, usually along the x- and y-axes. > Motions along the perpendicular motions are independent each other. > Accelerations in a given dimension only affect motions along that dimension. 12 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  13. Sample Problems 1. A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 m/s. The cliff is 50.0 m above a flat, horizontal beach as shown in the figure. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity? (c) Write the equations for the x - and y - components of the velocity of the stone with time. (d) Write the equations for the position of the stone with time, using the coordinates in the figure. (e) How long after being released does the stone strike the beach below the cliff? (f) With what speed and angle of impact does the stone land? (Serway) 13 Villacorta--FEU-Phys1BLec-L07-1819Sem02

  14. Sample Problems 2. A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started? (Serway) 14 Villacorta--FEU-Phys1BLec-L07-1819Sem02

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