Kinematics Kinematic Equations and Falling Objects
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1 Kinematics Kinematic Equations and Falling Objects Lana Sheridan De Anza College Sept 28, 2017
2 Last time kinematic quantities relating graphs
3 Overview derivation of kinematics equations using kinematics equations
4 Equations that describe motion in terms of position, velocity, acceleration, and time. To solve kinematics problems: identify what equations apply, then do some algebra to find the quantity you need.
5 For constant velocity (a = 0): x = vt (1)
6 For constant velocity (a = 0): x = vt (1) In general, when velocity is not constant, this is always true: x = v avg t
7 Always true: x = v avg t (2) This is just a rearrangement of the definition of average velocity: average velocity, v avg v avg = x t where x is the displacement that occurs in a time interval t.
8 If acceleration is constant (a = const): From v avg = x t, we also have: v = v 0 + at Note that Equation crosses the velocity ax graphical interpretatio I of Figure 2 9, the equ Also, note that m Equation 2 7 has the sa v avg = v i + v f 2 v v EXERCISE 2 2 A ball is thrown straight of the ball is m/s 2 a s, and b. Solution 1 v av = 2(v 0 + v) a. Substituting t = 0.5 (a) 1 Figure from James S. Walker Physics. v 0 O t t v = b. Similarly, using t = v =
9 Recall, x = v avg t and v avg = v i+v f 2 (constant acceleration) Substituting the second in to the first: ( vi + v f x = 2 ) t (3)
10 From the definition of acceleration: We can integrate this expression. a = dv dt
11 From the definition of acceleration: We can integrate this expression. For constant acceleration: a = dv dt v(t) = v i + at (4) where v i is the velocity at t = 0 and v(t) is the velocity at time t.
12 For constant acceleration: x(t) = x i + v i t at2 Equivalently, x = v i t at2 (5) How do we know this?
13 For constant acceleration: x(t) = x i + v i t at2 Equivalently, x = v i t at2 (5) How do we know this? Integrate!
14 The last equation we will derive is a scaler equation.
15 The last equation we will derive is a scaler equation. ( ) vi + v f x = t 2 We could also write this as: ( vi + v f ( x) i = 2 ) t i where x, v i, and v f could each be positive or negative.
16 The last equation we will derive is a scaler equation. ( ) vi + v f x = t 2 We could also write this as: ( vi + v f ( x) i = 2 ) t i where x, v i, and v f could each be positive or negative. We do the same for equation (4): v f i = (v i + at) i
17 The last equation we will derive is a scaler equation. ( ) vi + v f x = t 2 We could also write this as: ( vi + v f ( x) = 2 ) t where x, v i, and v f could each be positive or negative. We do the same for equation (4): v f = (v i + at) Rearranging for t: t = v f v i a
18 t = v f v i a ( vi + v f ; x = 2 Substituting for t in our x equation: x = ( ) ( ) vi + v f vf v i 2 a 2a x = (v i + v f )(v f v i ) so, ) t v 2 f = v 2 i + 2 a x (6)
19 For zero acceleration: Always: x = vt For constant acceleration: x = v avg t v f = v i + at x = v i t at2 x = v i + v f t 2 vf 2 = vi a x
20 Summary For zero acceleration: x = vt Always: x = v avg t For constant acceleration: v f = v i + at x = v i t at2 x = v i + v f t 2 vf 2 = vi a x
21 Using the Kinematics Equations Example Drag racers can have accelerations as high as 26.0 m/s 2. Starting from rest (v i = 0) with that acceleration, how much distance does the car cover in 5.56 s? Before we start answering...
22 Using the Kinematics Equations Process: 1 Identify which quantity we need to find and which ones we are given. 2 Is there a quantity that we are not given and are not asked for? 1 If so, use the equation that does not include that quantity. 2 If there is not, more that one kinematics equation may be required or there may be several equivalent approaches. 3 Input known quantities and solve.
23 Using the Kinematics Equations Example Drag racers can have accelerations as high as 26.0 m/s 2. Starting from rest (v i = 0) with that acceleration, how much distance does the car cover in 5.56 s? 1 OpenStax Physics
24 Using the Kinematics Equations Example Drag racers can have accelerations as high as 26.0 m/s 2. Starting from rest (v i = 0) with that acceleration, how much distance does the car cover in 5.56 s? What equation should we use? 1 OpenStax Physics
25 Using the Kinematics Equations Example Drag racers can have accelerations as high as 26.0 m/s 2. Starting from rest (v i = 0) with that acceleration, how much distance does the car cover in 5.56 s? What equation should we use? x = v i t at2 1 OpenStax Physics
26 Using the Kinematics Equations Example Drag racers can have accelerations as high as 26.0 m/s 2. Starting from rest (v i = 0) with that acceleration, how much distance does the car cover in 5.56 s? What equation should we use? x = v i t at2 x = 1 2 at2 x = 1 2 (26.0ms 2 )(5.56s) 2 = 402 m 1 OpenStax Physics
27 t will stop farther away from its starting point, so the answer to uation Using2.15 the that Kinematics if v Equations, Ex 2.8 xi is larger, then x f will be larger. imit! A car traveling at a constant speed of 45.0 m/s passes a trooper on a motorcycle hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch the car, accelerating at a constant rate of 3.00 AM m/s 2. How long does it take the trooper to overtake the car? sses a e secooper g at a ooper t 1.00 s t 0 t? y the er conunder Figure 2.13 (Example 2.8) A speeding car passes a hidden trooper. 1 Serway & Jewett, pg 39.
28 t will stop farther away from its starting point, so the answer to uation Using2.15 the that Kinematics if v Equations, Ex 2.8 xi is larger, then x f will be larger. imit! A car traveling at a constant speed of 45.0 m/s passes a trooper on a motorcycle hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets out from the billboard to catch the car, accelerating at a constant rate of 3.00 AM m/s 2. How long does it take the trooper to overtake the car? sses a e secooper g at a ooper t 1.00 s t 0 t? y the er conundeden Answer: Figure t = s(example 2.8) A speeding car passes a hid- trooper. 1 Serway & Jewett, pg 39.
29 Summary the kinematic equations using kinematics equations Collected Homework Due Friday, Oct 6. Quiz Tomorrow at the start of class. (Uncollected) Homework Serway & Jewett, Ch 2, onward from page 49. Conceptual Q: 7; Problems: 25, 29, 33, 39, 83
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