1.3 w
.pdf
School
University of Notre Dame**We aren't endorsed by this school
Course
PHYSICS 1
Subject
Physics
Date
Dec 16, 2024
Pages
4
Uploaded by SargentTeam17863
Circular and Projectile Motion If an object is moving along a curved path rather than a straight line, then the direction of the velocity vector is constantly changing. The change in direction constitutes an acceleration, regardless of whether or not the magnitude of the vector (the speed) is changing. Even if the speed isn't changing, as is frequently the case in circular motion, the change in direction means that the velocity is changing and that the object is accelerating. Since acceleration is defined as the rate of change of velocity, a = fi—‘; the acceleration may be visualized by drawing a vector subtraction diagram as shown below. SR 8 ——— ® If you start by writing the equation Av =v, —v;, you can turn it into a more familiar vector addition equation by thinking of subtraction as adding the opposite of v; such that Av = v, + (-v;). The diagram above shows how to visualize this calculation using the head-to-tail method. Notice that the vector Av is clearly not zero even though the speed is constant, and that Av points toward the center of the circle. I Develop and Use Models Draw a vector subtraction diagram above the projectile motion drawing at right showing that the change in velocity Av = v, — v, for a projectile points downward, in the direction of the acceleration due to gravity. 10 Investigation 1 Modeling Motion
Mathematical Practices: Reason Abstractly and Quantitatively During projectile motion, both the direction and the magnitude of the velocity vector are constantly changing. Because the horizontal and vertical components of projectile motion are independent, the arithmetic needs to be done on one component at a time. Note that there is no equation for the horizontal velocity component, v,, since it remains constant throughout the motion. Horizontal Equation of Motion Vertical Equations of Motion Ax = vt v, =V, + gt and Ay = vt + %gt2 Ax = horizontal displacement v, = vertical velocity component component V;, = initial vertical velocity component v, = horizontal velocity component Ay = vertical displacement component In projectile motion problems, you often need to use the given information with an equation for one component to generate a value for the time an object has been moving, and then use that time in an equation for the other component to calculate the quantity of interest. For example, you might do an experiment where a marble rolls off a table that is 0.85 m tall and hits the floor at a distance of 0.66 m from the edge of the table. You can use this information to determine the initial speed of the marble as follows. First use the vertical position-time equation to calculate the time needed for the marble to reach the floor. Since the table is horizontal, you can assume the initial vertical velocity component is zero and solve for time: Ay = (0 m/s)t + 1g* = 1g7’ 7 _ 2Ay = e [2Ay _ [2(=0.85m) B g ¥V —9.8m/s? r=042s Then use this time in the horizontal position-time equation to solve for the horizontal velocity: Ax=vt Circular and Projectile Motion 11
3-Dimensioncal Review 1. DCI Forces and Motion A 14 kg boulder is pushed off a cliff with velocity v = (14.0 m/s)X + (2.0 m/s)y. Will the object experience a larger vertical or horizontal acceleration? Explain your answer. After being pushed off the cliff, the boulder’s horizontal motion will be uniform so the horizontal acceleration will be zero. The boulder’s vertical motion will have a negative acceleration due to gravity. So although the horizontal acceleration is zero, it is larger than the vertical acceleration. 2. SEP Analyzing and Interpreting Data A golfer hits a ball into the air with initial velocity v =v, + v, a. Is the horizontal motion of the golf ball uniform? Explain your answer. The horizontal motion of the ball is uniform because there is zero acceleration in the horizontal direction. b. Is the vertical motion of the ball uniform? Explain your answer. As the ball is in the air, it experiences the vertical pull of gravity. Thus the vertical motion of the ball is not uniform because it has a negative acceleration due to gravity. c. Fill in the position vs. time, velocity vs. time, and acceleration vs. time graphs for the horizontal and vertical motion of the golf ball. 3. CCC Cause and Effect A projectile is shoved horizontally off a cliff. Which of the following would cause the projectile to have a farther horizontal displacement before hitting the ground? a greater initial horizontal velocity b. a greater initial vertical velocity c. a greater projectile mass d. a shorter cliff 12 Investigation 1 Modeling Motion
Skills Practice 4. A professional football punter kicks a football with an initial velocity v = (14.0 m/s)% + (21.0 m/s)y. How long the football stays in the air is known as the hang time. Determine the hang time, as well as the horizontal and maximum vertical displacements. 4 t=429s5,Ax=60.1m, Ay=22.6m 5. A professional golfer hits a ball with an initial velocity v = (19.0 m/s)z + (26.0 m/s)$. How long the golf ball stays in the air is known as the hang time. Determine the hang time, as well as the horizontal and maximum vertical displacements. 4 t=531s,Ax=101m, Ay=345m 6. A professional football punter kicks a football with an initial velocity v = (16.0 m/s)£ + (23.0 m/s)9. Determine the horizontal and maximum vertical displacements. a.Ax=376m, Ay=29.8m b. Ax=37.6m, Ay=39.5m @Ax =750m, Ay=270m d. Ax=750m, Ay=395m Circular and Projectile Motion 13