# Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 4: Motion with a Changing Velocity.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 4: Motion with a Changing Velocity Motion Along a Line Graphical Representation of Motion Free Fall Projectile Motion Apparent Weight Air Resistance and Terminal Velocity

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 4.1 Motion Along a Line For constant acceleration the kinematic equations are: Also:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Example: In a previous example, a box sliding across a rough surface was found to have an acceleration of -2.94 m/s 2. If the initial speed of the box is 10.0 m/s, how long does it take for the box to come to rest? Know: a= -2.94 m/s 2, v ix =10.0 m/s, v fx = 0.0 m/s Want:  t.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Example (text problem 4.8): A train of mass 55,200 kg is traveling along a straight, level track at 26.8 m/s. Suddenly the engineer sees a truck stalled on the tracks 184 m ahead. If the maximum possible braking force has magnitude 84.0 kN, can the train be stopped in time? Know: v fx = 0 m/s, v ix =26.8 m/s,  x=184 m Determine a x and compare to the train’s maximum a x.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 Example continued: The train’s maximum acceleration is: The maximum acceleration is not sufficient to stop the train before it hits the stalled truck.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 § 4.2 Visualizing Motion with Constant Acceleration Motion diagrams for three carts:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 Graphs of x, v x, a x for each of the three carts

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 Example (text problem 4.13): A trolley car in New Orleans starts from rest at the St. Charles Street stop and has a constant acceleration of 1.20 m/s 2 for 12.0 seconds. (a) Draw a graph of v x versus t.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 (b) How far has the train traveled at the end of the 12.0 seconds? The area between the curve and the time axis represents the distance traveled. (c) What is the speed of the train at the end of the 12.0 s? This can be read directly from the graph, v x =14.4 m/s. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 § 4.3 Free Fall A stone is dropped from the edge of a cliff; if air resistance can be ignored, the FBD for the stone is: x y w Apply Newton’s Second Law The stone is in free fall; only the force of gravity acts on the stone.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise? Given: v iy =+15.0 m/s; a y =-9.8 m/s 2 x y v iy ayay To calculate the final height, we need to know the time of flight. Time of flight from:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 The ball rises until v fy = 0. The height: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Example (text problem 4.22): A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. 369 m x y Given: v iy =0 m/s, a y =-9.8 m/s 2,  y=-369 m Unknown: v yf Use: ayay

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 How long does it take for the penny to strike the ground? (downward) Example continued: Given: v iy =0 m/s, a y =-9.8 m/s 2,  y=-369 m Unknown:  t

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 §4.4 Projectile Motion What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball. The baseball has a x = 0 and a y = -g, it moves with constant velocity along the x-axis and with nonzero, constant acceleration along the y-axis.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 Example: An object is projected from the origin. The initial velocity components are v ix = 7.07 m/s, and v iy = 7.07 m/s. Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results. Since the object starts from the origin,  y and  x will represent the location of the object at time  t.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 t (sec)x (meters)y (meters) 000 0.21.411.22 0.42.832.04 0.64.242.48 0.85.662.52 1.07.072.17 1.28.481.43 1.49.890.29 Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 This is a plot of the x position (black points) and y position (red points) of the object as a function of time. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 Example continued: This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Example (text problem 4.36): An arrow is shot into the air with  = 60° and v i = 20.0 m/s. (a) What are v x and v y of the arrow when t=3 sec? x y 60° The components of the initial velocity are: At t = 3 sec:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 (b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval? y x r Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 Example: How far does the arrow in the previous example land from where it is released? The arrow lands when  y=0. Solving for  t: The distance traveled is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 §4.5 Apparent Weight Stand on a bathroom scale. FBD: Apply Newton’s 2 nd Law: w N x y

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Example: A woman of mass 51 kg is standing in an elevator. If the elevator pushes up on her feet with 408 newtons of force, what is the acceleration of the elevator? FBD for woman: Apply Newton’s 2 nd Law: (1) w N x y

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Solving (1) for a y : The elevator could be (1) traveling upward with decreasing speed, or (2) traveling downward with increasing speed. Given: N = 408 newtons, m = 51 kg, g = 9.8 m/s 2 Unknown: a y Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 §4.6 Air Resistance A stone is dropped from the edge of a cliff; if air resistance cannot be ignored, the FBD for the stone is: Apply Newton’s Second Law x y w FdFd Where F d is the magnitude of the drag force on the stone. This force is directed opposite the object’s velocity

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Assume that b is a parameter that depends on the size and shape of the object. Since F d  v 2, can the object be in equilibrium?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 Example (text problem 4.61): A paratrooper with a fully loaded pack has a mass of 120 kg. The force due to air resistance has a magnitude of F d = bv 2 where b = 0.14 N s 2 /m 2. (a) If he/she falls with a speed of 64 m/s, what is the force of air resistance?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 (b) What is the paratrooper’s acceleration? Apply Newton’s Second Law and solve for a. x y w FdFd FBD: (c) What is the paratrooper’s terminal speed? Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Summary The Kinematic Equations Graphical Representations of Motion Applications of Newton’s Second Law & Kinematics (free fall, projectiles, accelerated motion, air drag) Terminal Velocity

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