EQ: EQ: Acceleration Due to Gravity How How is is the the motion motion of of an an object object moving moving vertically vertically different different from from one one moving moving horizontally? horizontally?
fall. fall. Acceleration Due to Gravity Every Every object object on on the the earth earth experiences experiences aa common common force: force: the the force force due due to to gravity. gravity. This This force force is is always always
directed directed toward toward the the center center of of the the earth earth (downward). (downward). Free Free fall fall can can be be analyzed analyzed using using the the same same kinematic kinematic equations equations we
we used used to to analyze analyze horizontal horizontal motion. motion. But But there there are are aa few few factors factors Gravitational Acceleration InIn aa vacuum, vacuum, all all objects objects fall fall with with same same acceleration. acceleration. Equations
Equations for for constant constant acceleration acceleration apply apply as as usual. usual. Near Near the the Earths Earths surface: surface: a = g = - 9.80 m/s2 Directed downward (always negative). The Acceleration of Gravity (g) If two rocks with different mass are dropped (vi = 0) from the same height at the same time, would one land before the other? Or would
they land at the same time? Galileo demonstrated that g is the same for all objects, regardless of their mass! This was confirmed by the Apollo astronauts on the Moon, where there is no air resistance http:// www.youtube.com/watch?v=7eTw35Z D1Ig Falling objects accelerate at a constant rate (Galileo): All objects (regardless of their mass) experience the same acceleration when free falling. No air resistance present. When the only force is gravity, the acceleration is the same for all objects. On Earth, this acceleration (g) is 9.8 m/s2.
When an object is dropped or simple falls, it will have an vi = 0 m/s since no initial force was applied to make it move. Gravitational Acceleration When an object is thrown straight up, it will have a velocity of 0 m/s2 at its peak height just before it begins its free fall. This means that for a split second, an object will stop in mid air while it transitions from upward motion to downward motion. Because gravity is constant, the rate at which an object decelerates as it moves upward, is the same rate it accelerates at as it falls downward. In other words, an object falling will have the same speed at any point that it did on the way up. 6 Various Types of Free Fall 0 m/s 9.8 m/s
19.6 m/s 29.4 m/s 39.2 m/s 49.0 m/s 7 Graphing Free Fall Motion Talk out your misconceptions about free falling objects with a shoulder partner then at the bottom of your notes explain: -Why any 2 objects will hit the ground first regardless of their mass (in the absence of air resistance) -Why the initial velocity of free falling objects is zero -What direction does gravity act on (side to side or up and down) -What does it mean to say that the acceleration of gravity is 9.80 m/s2 (for ex: what is happening to the object every second as it falls) Sign Convention: A Ball Thrown Vertically Upward
x= + v= 0 x= + v= + UP = + Release Point x= + v= x= 0 v= - x= v= - Displacement Displacement is is positive positive (+) (+) or or negative negative (-) (-) based based on on LOCATION.
LOCATION. Velocity Velocity is is positive positive (+) (+) or or negative negative (-) (-) based based on on direction direction of of motion. motion. Acceleration is gravity which is always downward (negative). Kinematic Equations Horizontal Motion v d t
Vertical Motion v d t vf = vi + a(t)t)t) vf = vi + g(t)t)t) x = (vi + vf)t)t y = (vi + vf)t)t x = vi(t)t)t) + a(t)t)t)2 y = vi(t)t)t) + g(t)t)t)2 vf2 = vi2 + 2at)x vf2 = vi2 + 2gt)y Drop/fallVi=0 m/s Height, tall y g= acceleration due
to gravity = -9.81 m/s2 Same Problem Solving Strategy Except a = g: Draw and label sketch of problem. Indicate + direction. List givens and state what is to be found. Select equation containing one and not the other of the unknown quantities, and solve for the unknown. Dont Forget Units!!!!! Example 7: A ball is thrown vertically upward with an initial velocity of 30 m/s. What are its position and velocity after 2 s, 4 s, and 7 s? Step 1. Draw and label a sketch. Step 2. Indicate + direction. Step 3. Identify given variables.
a = -9.8 ft/s2 s t = 2, 4, 7 vi = + 30 m/s y= ? v + a= g vi = +30 m/s Finding Displacement: Step 4. Select equation that contains x and not v. y vi t x = (30)t + 1 2
at 2 + a= g (-9.8)t2 Substitution of t = 2, 4, and 7 s will give the following values: vo = 30 m/s yy = = 40.4 40.4 m; m; y= y= 41.6 41.6 m; m; y= y= -30.1 -30.1
m m Finding Velocity: Step Step 5. 5. Find Find vv from from equation equation that that contains contains vv and and not not x: x: v f vi at + a= g v f (30) ( 9.8)t Substitute t = 2, 4, and 7
s: vo = 30 m/s vv = = +10.4 +10.4 m/s; m/s; vv = = -9.20 -9.20 m/s; m/s; vv = = -38.6 -38.6 m/s m/s Example 7: (Cont.) Now find the maximum height attained: Now find an equation that uses y but not t. Keep in mind the velocity at its highest point is zero 2 +
a= g 2 v f vi 2 gy 2 2 0 (30) 2( 9.8) y 2 30 y 45.9m 2( 9.8) vo = +30 m/ s The tallest Sequoia sempervirens tree in Californias Redwood National Park is 111 m tall. Suppose an object is
thrown downward from the top of that tree with a certain initial velocity. If the object reaches the ground in 3.80 s, what is the Kinematic Equations Horizontal Motion v d t Vertical Motion v d t vf = vi + a(t)t)t) vf = vi + g(t)t)t) x = (vi + vf)t)t
y = (vi + vf)t)t x = vi(t)t)t) + a(t)t)t)2 y = vi(t)t)t) + g(t)t)t)2 vf2 = vi2 + 2at)x vf2 = vi2 + 2gt)y Drop/fallVi=0 m/s Height, tall y g= acceleration due to gravity = -9.81 m/s2 G: E: S: U: vi S:
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Wen Ming Liu (Concordia University) Joint work with: LingyuWang (Concordia University) KuiRen (University at Buffalo) MouradDebbabi (Concordia University) Cloudcom 2013. [email protected] / [email protected] December 4 , 2013. Concordia Institute for Information Systems Engineering (CIISE) at Concordia University
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