CH 03_PHY

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10/28/2023 1 Lecture and discussions PHYS 1401 College Physics Two-Dimensional Kinematics Two-Dimensional Kinematics Department of Physical Sciences South Texas College
10/28/2023 2 Lecture and discussions PHYS 1401 College Physics This OpenStax ancillary resource is © Rice University under a CC-BY 4.0 International license; it may be reproduced or modified but must be attributed to OpenStax, Rice University and any changes must be noted.
10/28/2023 3 Lecture and discussions PHYS 1401 College Physics Chapter Outline 3. 1 Kinematic s i n T w o Dimensions : A n Intr oduc tion • Observe that motion in two dimensions consists of horizontal and vertical components. • Understand the independence of horizontal and vertical vectors in two-dimensional motion. 3.2 Vector Addition and Subtraction: Graphical Methods • Understand the rules of vector addition, subtraction, and multiplication. • Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects. 3.3 Vector Addition and Subtraction: Analytical Methods • Understand the rules of vector addition and subtraction using analytical methods. • Apply analytical methods to determine vertical and horizontal component vectors. • Apply analytical methods to determine the magnitude and direction of a resultant vector.
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10/28/2023 4 Lecture and discussions PHYS 1401 College Physics 3. 4 P r ojec til e M otion • Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory. • Determine the location and velocity of a projectile at different points in its trajectory. • Apply the principle of independence of motion to solve projectile motion problems. 3. 5 Additio n o f V elocities • Apply principles of vector addition to determine relative velocity. • Explain the significance of the observer in the measurement of velocity.
10/28/2023 5 Lecture and discussions PHYS 1401 College Physics
Section 3.1 PHYS 1401 College Physics 10/28/2023 6 Lecture and discussions Coordinate system • Cartesian Coordinate System Uses horizontal axis (X- axis) and vertical axis ( Y-axis) in two dimension Represented by (x , y) • Polar Coordinate System Uses distance from a reference point and angle from a reference direction Represented by ( r, θ ) • Conversion of cartesian coordinate system to polar coordinate system • X 2 +y 2 =r 2 • θ = tan -1 (y/x) • X= r cosθ, Y = r sinθ sin y r cos x r
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Polar to Cartesian Coordinates Section 3.1 PHYS 1401 College Physics Polar coordinate system (r,θ ) Cartesian Coordinate system ( x, y) Conversion of cartesian to polar r 2 =x 2 +y 2
10/28/2023 8 Lecture and discussions PHYS 1401 College Physics • A pedestrian walks a two-dimensional path between two points in a city. In this scene, all blocks are square and are the same size. A A x A y ᶿ
10/28/2023 9 Lecture and discussions PHYS 1401 College Physics
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10/28/2023 10 Lecture and discussions PHYS 1401 College Physics • To describe the resultant vector for the person walking in a city considered in Figure 3.9 graphically, draw an arrow to represent the total displacement vector D . • Using a protractor, draw a line at an angle θ relative to the east-west axis. The length D of the arrow is proportional to the vector’s magnitude and is measured along the line with a ruler. In this example, the magnitude D of the vector is 10.3 units, and the direction θ is 29.1º north of east. Scaling : Say, displacement is 67 miles, angle is 30 0, how you draw the displacement? May be, 1 mile = 0.1 cm?, 67 miles = 6.7 cm??, how do you determine the scaling?
10/28/2023 11 Lecture and discussions PHYS 1401 College Physics • The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So B is the negative of –B ; it has the same length but opposite direction.
10/28/2023 12 Lecture and discussions PHYS 1401 College Physics Suppose, for example, that is the vector representing the total displacement of the person walking in a city Can be expressed in the following way: A x = ? A y = ? = ? ᶿ
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10/28/2023 13 Lecture and discussions PHYS 1401 College Physics Suppose, for example, that is the vector representing the total displacement of the person walking in a city Can be expressed in the following way:
10/28/2023 14 Lecture and discussions PHYS 1401 College Physics • This shows the motions of two identical balls—one falls from rest, the other has an initial horizontal velocity. Each subsequent position is an equal time interval. • Arrows represent horizontal and vertical velocities at each position. The ball on the right has an initial horizontal velocity, while the ball on the left has no horizontal velocity. • Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls. This shows that the vertical and horizontal motions are independent.
PHYS 1401 College Physics Adding Vectors • Vector addition is very different from adding scalar quantities. • When adding vectors, their directions must be taken into account. • Units must be the same • Graphical Methods – Use scale drawings • Algebraic Methods – More convenient
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• To add vectors A and B , first determine the horizontal and vertical components of each vector. These are the dotted vectors A x , A y , B x and B y shown in the image. PHYS 1401 College Physics
10/28/2023 17 Lecture and discussions PHYS 1401 College Physics
• The magnitude of the vectors A x and B x add to give the magnitude R x of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors A y and B y add to give the magnitude R y of the resultant vector in the vertical direction. PHYS 1401 College Physics
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• Using analytical methods, we see that the magnitude of R is 81.2 m and its direction is 36.6º north of east. PHYS 1401 College Physics
• The subtraction of the two vectors shown in Figure 3.30 . The components of –B are the negatives of the components of B . The method of subtraction is the same as that for addition. PHYS 1401 College Physics
10/28/2023 21 Lecture and discussions PHYS 1401 College Physics Add the vector to the vector shown in Figure 3.33, using perpendicular components along the x- and y-axes. The x- and y-axes are along the east–west and north–south directions, respectively. Vector A represents the first leg of a walk in which a person walks in a direction north of east. Vector B represents the second leg, a displacement of 34 m in a direction north of east
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10/28/2023 22 Lecture and discussions PHYS 1401 College Physics Add the vector to the vector shown in Figure 3.33, using perpendicular components along the x- and y-axes. The x- and y-axes are along the east–west and north–south directions, respectively. Vector A represents the first leg of a walk in which a person walks in a direction north of east. Vector B represents the second leg, a displacement of 34 m in a direction north of east
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10/28/2023 23 Lecture and discussions PHYS 1401 College Physics
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PHYS 1401 College Physics Projectile motion Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory http://ffden-2.phys.uaf.edu/webproj/211_fall_2014/Stephen_Ringle/stephen_ringle/ProjectileMotion.html
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PHYS 1401 College Physics Projectile motion http://ffden-2.phys.uaf.edu/webproj/211_fall_2014/Stephen_Ringle/stephen_ringle/ProjectileMotion.html
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10/28/2023 26 Lecture and discussions PHYS 1401 College Physics
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• The total displacement s of a soccer ball at a point along its path. The vector s has components x and y along the horizontal and vertical axes. Its magnitude is s , and it makes an angle θ with the horizontal. PHYS 1401 College Physics
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(a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because a x = 0 and v x is thus constant. (c) The velocity in the vertical direction begins to decrease as the object rises; at its highest point, the vertical velocity is zero. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The x - and y -motions are recombined to give the total velocity at any given point on the trajectory. PHYS 1401 College Physics
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• The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of 233 m and 125 m away horizontally. PHYS 1401 College Physics
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• The trajectory of a rock ejected from the Kilauea volcano. PHYS 1401 College Physics
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• Trajectories of projectiles on level ground. (a) The greater the initial speed v 0 , the greater the range for a given initial angle. (b) The effect of initial angle θ 0 on the range of a projectile with a given initial speed. Note that the range is the same for 15º and 75º , although the maximum heights of those paths are different. PHYS 1401 College Physics
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PHYS 1401 College Physics Projectile motion Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory j/211_fall_2014/Stephen_Ringle/stephen_ringle/ProjectileMotion.html Maximum height (H max ), horizontal range R , time of flight (T)
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PHYS 1401 College Physics Projectile motion Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory j/211_fall_2014/Stephen_Ringle/stephen_ringle/ProjectileMotion.html Maximum height (H max ), horizontal range R , time of flight (T)
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• The man is walking on the moving beltway. • The woman on the beltway sees the man walking at his normal walking speed. • The stationary woman sees the man walking at a much higher speed. • The difference is due to the relative velocity of their frames of reference. Section 4.6 PHYS 1401 College Physics Relative Velocity
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Relative Velocity Section 4.6 S A is stationary S B is moving The frame S B is moving with velocity which means, velocity of B, with respect to A PHYS 1401 College Physics BA v PA PB BA t r r v PA PB BA u u v
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10/28/2023 36 Lecture and discussions PHYS 1401 College Physics During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of above the horizontal. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?
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10/28/2023 37 Lecture and discussions PHYS 1401 College Physics During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of above the horizontal, as illustrated in Figure. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?
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10/28/2023 38 Lecture and discussions PHYS 1401 College Physics Calculating Projectile Motion: Hot Rock Projectile Kilauea in Hawaii is the world’s most continuously active volcano. Very active volcanoes characteristically eject red-hot rocks and lava rather than smoke and ash. Suppose a large rock is ejected from the volcano with a speed of 25.0 m/s and at an angle above the horizontal, as shown in Figure 3.39. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path. (b) What are the magnitude and direction of the rock’s velocity at impact?
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10/28/2023 39 Lecture and discussions PHYS 1401 College Physics Calculating Projectile Motion: Hot Rock Projectile Kilauea in Hawaii is the world’s most continuously active volcano. Very active volcanoes characteristically eject red-hot rocks and lava rather than smoke and ash. Suppose a large rock is ejected from the volcano with a speed of 25.0 m/s and at an angle above the horizontal, as shown in Figure 3.39. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path. (b) What are the magnitude and direction of the rock’s velocity at impact?
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10/28/2023 40 Lecture and discussions PHYS 1401 College Physics
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10/28/2023 41 Lecture and discussions PHYS 1401 College Physics Rain is falling vertically with a speed of 35m/s. A women rides a bicycle with a speed of 12 m/s in east to west direction. In which direction should she hold her umbrella? N s V RE V WE V EW V RW
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10/28/2023 42 Lecture and discussions PHYS 1401 College Physics Rain is falling vertically with a speed of 35m/s. A women rides a bicycle with a speed of 12 m/s in east to west direction. In which direction should she hold her umbrella? the magnitude of relative velocity the rain drops wrt woman = √(12² + 35²) = 37 m/s direction of relative velocity of the rain drops = = tan ¹ (12/35) = 18.92 deg. from the ⁻ vertical. So the cyclist should hold the umbrella at 18.92 deg to the vertical in the forward direction.
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10/28/2023 43 Lecture and discussions PHYS 1401 College Physics Calculating Relative Velocity: An Airline Passenger Drops a Coin An airline passenger drops a coin while the plane is moving at 260 m/s. What is the velocity of the coin when it strikes the floor 1.50 m below its point of release: (a) Measured relative to the plane? (b) Measured relative to the Earth? The motion of a coin dropped inside an airplane as viewed by two different observers. (a) An observer in the plane sees the coin fall straight down. (b) An observer on the ground sees the coin move almost horizontally.
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10/28/2023 44 Lecture and discussions PHYS 1401 College Physics Refer to Figure 3.46, which shows a boat trying to go straight across the river. Let us calculate the magnitude and direction of the boat’s velocity relative to an observer on the shore, V tot . The velocity of the boat,V bot , is 0.75 m/s in the -direction relative to the river and the velocity of the river, V river , is 1.20 m/s to the right. Figure 3.46 A boat attempts to travel straight across a river at a speed 0.75 m/s. The current in the river, however, flows at a speed of 1.20 m/s to the right.
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